1 Statement of the results and motivation
1.1 Main results
Consider a subshift of finite type $({\mathbb X}, T)$ endowed with a Gibbs measure $\nu $ and let f be a real-valued Hölder continuous function on ${\mathbb X}$ (the precise definitions are given in §2). Define the Birkhoff sums
A fundamental result of the theory of dynamical systems is the celebrated Birkhoff ergodic theorem which asserts that $\nu $ -almost surely,
Much effort was made to establish another important property: the central limit theorem for $S_nf$ . To formulate the corresponding statement, we first note that the following limit exists:
It is known that $\sigma _f^2=0$ if and only if f is a coboundary with respect to T, which means that there exists a Hölder continuous function g on ${\mathbb X}$ such that $f(x) = g(T x) - g(x)$ for any $x \in {\mathbb X}$ . In the case when $\sigma _{f }>0$ (or, equivalently, when f is not a coboundary) the following central limit theorem holds: for any bounded continuous function $F: {\mathbb R} \mapsto {\mathbb R}$ ,
All these statements, which can be found in the excellent book of Parry and Pollicott [Reference Parry and Pollicott24], are consequences of many successive works such as Sinai [Reference Sinai27, Reference Sinai28], Ratner [Reference Ratner25], Ruelle [Reference Ruelle26] and Denker and Phillip [Reference Denker and Philipp9], to cite only a few. The goal of this paper is to complement the central limit theorem (1.1) by proving limit theorems for the Birkhoff sum $t+S_nf$ under the condition that the trajectory $(t+S_kf)_{1\leqslant k\leqslant n}$ stays positive, where $t\in {\mathbb R}$ is a starting point.
There is a vast body of literature on the properties of conditioned random walks and their applications based on independent observations: a brief historical foray into the subject of conditioned limit theorems and our motivation are presented in §1.2. At this point let us note that finding the corresponding asymptotics for Birkhoff sums encounters major problems. One of them is related to the fact that Wiener–Hopf factorization techniques do not apply in these settings. The other, and this is one of the main findings of the paper, is that the asymptotic analysis requires the introduction of the new object, the harmonic measure, which makes an important difference with the case of simple random walks. Regarding potential applications, we note that counting for trajectories conditioned to stay in some conic domains of ${\mathbb R}^d$ (for instance, the Weyl chamber) is of interest in statistical physics, see Fisher [Reference Fisher14]. Our study which deals with the one-dimensional observable f is the first step in considering such problems, and open ways to cope also with observables taking values in ${\mathbb R}^d$ .
To state our results assume that $\nu (f) = 0$ and that f is not a coboundary. For any $t \in {\mathbb R}$ , the following exit time is finite for $\nu $ -almost every $x \in {\mathbb X}$ :
Thus, by definition, $\{x \in {\mathbb X}: \tau _t^f(x)>n\}$ is the set where the trajectory $(t+S_kf)_{1\leqslant k\leqslant n}$ stays non-negative, that is, $t+S_kf \geqslant 0$ for $1\leqslant k\leqslant n$ .
Our first theorem states the existence of a special Radon measure which will play a central role in the paper and will be used in the formulations of the subsequent results.
Theorem 1.1. Let f be a Hölder continuous function on ${\mathbb X}$ such that $\nu (f) = 0$ and f is not a coboundary. Then, there exists a unique Radon measure $\mu ^f$ on ${\mathbb X} \times {\mathbb R}$ such that for any continuous compactly supported function $\varphi $ on ${\mathbb X} \times {\mathbb R}$ ,
Moreover, the Radon measure $\mu ^f$ satisfies the following quasi-invariance property: for every continuous compactly supported function $\varphi $ on ${\mathbb X} \times {\mathbb R}$ ,
The limit (1.2) takes a simpler form when the function $\varphi $ does not depend on the first argument. Indeed, we show in §3.3 that the marginal of $\mu ^f$ on ${\mathbb R}$ is absolutely continuous with respect to the Lebesgue measure. Its density function is a non-decreasing function on ${\mathbb R}$ that will be denoted by $V^f$ . In particular, by standard arguments, the asymptotic (1.2) is valid for functions $\varphi $ of the form for $x \in {\mathbb X}$ and $t \in {\mathbb R}$ . This leads to the following.
Corollary 1.2. Let f be a Hölder continuous function on ${\mathbb X}$ such that $\nu (f) = 0$ and f is not a coboundary. Then, for any real numbers $-\infty < a < b < \infty $ , we have
Note that (1.2) and (1.4) are stated in integral forms with respect to t. It is an open question whether it is possible to give an asymptotic of the integral for a fixed value of t.
The Radon measure $\mu ^f$ appearing in Theorem 1.1 will be called the harmonic measure associated to the dynamical system $({\mathbb X}, T, \nu )$ with the observable f. The reason for this is that the measure $\mu ^f$ is related to the harmonicity property that appears in the study of killed random walks on the half line. We refer to §3.3 for precise statements.
The following results describe the limit behavior of the Birkhoff sum $t + S_n f$ under the condition that the trajectory $(t + S_k f)_{1 \leqslant k \leqslant n}$ stays non-negative. We start by giving the equivalent of the probability that the trajectory $(t + S_k f)_{1 \leqslant k \leqslant n}$ stays non-negative. Denote by $\check {\mu }^f$ the harmonic measure related to the reversed dynamical system $({\mathbb X}, T^{-1}, \nu )$ with the observable $f \circ T^{-1}$ .
Theorem 1.3. Let f be a Hölder continuous function on ${\mathbb X}$ such that f is not a coboundary and $\nu (f) = 0$ . Then, for any continuous compactly supported function $\varphi $ on ${\mathbb X} \times {\mathbb R}$ , we have
and
As the measure $\mu ^f$ has absolutely continuous marginal on ${\mathbb R}$ , Theorem 1.3 also applies to the function for $x \in {\mathbb X}$ and $t \in {\mathbb R}$ . In particular, this gives the following corollary.
Corollary 1.4. Let f be a Hölder continuous function on ${\mathbb X}$ such that f is not a coboundary and $\nu (f) = 0$ . Then, for any real numbers $-\infty < a < b < \infty $ , we have
Now we give a conditioned central limit theorem for the Birkhoff sum $S_n f$ , which states that the law of $S_n f$ conditioned to stay positive converges weakly to the Rayleigh law. In the following, we denote by $\phi ^+$ and $\Phi ^+$ the Rayleigh density and cumulative distribution functions, respectively:
Theorem 1.5. Let f be a Hölder continuous function on ${\mathbb X}$ such that f is not a coboundary and $\nu (f) = 0$ . Then, for any continuous compactly supported function F on ${\mathbb X} \times {\mathbb X} \times {\mathbb R} \times {\mathbb R}$ , we have
As previously, we can actually apply Theorem 1.5 to the function for $x, x' \in {\mathbb X}$ and $t, t' \in {\mathbb R}$ . Therefore, this implies the following corollary.
Corollary 1.6. Let f be a Hölder continuous function on ${\mathbb X}$ such that f is not a coboundary and $\nu (f) = 0$ . Then, for any real numbers $-\infty < a < b < \infty $ and $-\infty < a' < b' < \infty $ , we have
Next we formulate a conditioned local limit theorem for $S_n f$ , which is a refinement of the previous result.
Theorem 1.7. Let f be a Hölder continuous function on ${\mathbb X}$ such that $\nu (f) = 0$ . Assume that for any $p \neq 0$ and ${q \in {\mathbb R}}$ , the function $p f + q$ is not cohomologous to a function with values in ${\mathbb Z}$ . Then, for any continuous compactly supported function F on ${\mathbb X} \times {\mathbb X} \times {\mathbb R} \times {\mathbb R}$ , we have
In Theorem 1.7, we assumed that the function f satisfies a non-arithmeticity condition. When this is not the case but f is still not cohomologous to $0$ , we could still get an analogue of this result by the same method.
In the particular case when the function F has the form for $x, x' \in {\mathbb X}$ and $t, t' \in {\mathbb R}$ , from the previous theorem we obtain the following.
Corollary 1.8. Let f be a Hölder continuous function on ${\mathbb X}$ such that $\nu (f) = 0$ . Assume that for any $p \neq 0$ and $q \in {\mathbb R}$ , the function $p f + q$ is not cohomologous to a function with values in ${\mathbb Z}$ . Then, for any real numbers $-\infty < a < b < \infty $ and $-\infty < a' < b' < \infty $ , we have
In this corollary we have denoted by $\check V^{(-f)}$ the density function with respect to the Lebesgue measure of the marginal on ${\mathbb R}$ of the Radon measure $\check {\mu }^{(-f)}$ .
From Corollary 1.8 we get a local limit theorem for $\tau _t^f(x)$ . Indeed, by taking the interval $[a', b')$ to be $[-c, 0)$ for $c>0$ large enough, as f is bounded, we deduce the following.
Corollary 1.9. Let f be a Hölder continuous function on ${\mathbb X}$ such that $\nu (f) = 0$ . Assume that for any $p \neq 0$ and $q \in {\mathbb R}$ , the function $p f + q$ is not cohomologous to a function with values in ${\mathbb Z}$ . Then, for any real numbers $-\infty < a < b < \infty $ , we have
Our Corollary 1.9 could be extended without difficulties to the case when one only assumes that f is not cohomologous to $0$ . This assertion could be deduced from a version of Theorem 1.7 for functions f that are cohomologous to functions with values in a set of the form $\alpha {\mathbb Z} + \beta $ for some $\alpha , \beta \in {\mathbb R}$ .
Similarly to the comment after Corollary 1.2, Theorems 1.3, 1.5 and 1.7 are stated in integral forms with respect to t. It is an open problem to obtain asymptotics for a fixed value of $t \in {\mathbb R}$ of the following probabilities:
Remark 1.10. In the previous theorems we have considered the two-sided subshift. However, all the above results apply as well to the case of one-sided subshift. The latter is a particular case of the two-sided one with a function f depending only on the future (or on the past). Indeed, let $({\mathbb X}^+, T)$ be the one-sided shift associated with ${\mathbb X}$ , $\psi :{\mathbb X}^+\rightarrow {\mathbb R}$ be the potential of the Gibbs measures $\nu $ and the function f only depends on the future coordinates in ${\mathbb X}$ . Then, for instance, the conclusions (1.2) and (1.3) of Theorem 1.1 may be rewritten as follows: for any continuous compactly supported function $\varphi $ on ${\mathbb X}^+\times {\mathbb R}$ ,
and
where $\nu ^+$ is the marginal of $\nu $ and $\mu _+^f$ is the marginal of $\mu ^f$ on ${\mathbb X}^+\times {\mathbb R}$ . In Theorem 1.3, in the case of a one-sided shift, the limit in the right-hand side of (1.6) exists. Nevertheless, even if the function f depends only on future coordinates, in order to construct the marginal of the measure $\check {\mu }^{(-f)}$ , we need to work in the full shift ${\mathbb X}$ and to apply Theorem 1.1 to the inverse map $T^{-1}$ . In the same way, in Theorem 1.7, the left-hand side of (1.8) makes sense in a one-sided shift, but we need to use the two-sided shift in order to make sense of the right-hand side. We refer to §§2.1 and 2.2 for more details about the relation between one-sided and two-sided subshifts.
Due to the theory of Markov partitions (see Appendix III of [Reference Parry and Pollicott24] and Ch. 18.7 of [Reference Hasselblatt and Katok20]), Theorems 1.1, 1.3, 1.5 and 1.7 can be applied without any changes to hyperbolic dynamical systems. Finally, using the approach of this paper, one can obtain analogous results for hyperbolic flows. The latter is beyond the scope of this article and will be done in another work.
1.2 Previous work and motivation
The first examples of conditioned limit theorems for sums of independent random variables are due to the pioneering work of Spitzer [Reference Spitzer29] and Feller [Reference Feller13]. Since then integral and local limit theorems for random walks conditioned to stay positive attracted a lot of attention. Very many authors contributed to this subject, among them Borovkov [Reference Borovkov3–Reference Borovkov5], Bolthausen [Reference Bolthausen2], Iglehart [Reference Iglehart22], Eppel [Reference Eppel12], Bertoin and Doney [Reference Bertoin and Doney1], Caravenna [Reference Caravenna6], Vatutin and Wachtel [Reference Vatutin and Wachtel32], Doney [Reference Doney10] and Kersting and Vatutin [Reference Kersting and Vatutin23]. Most of this work is based on the Wiener–Hopf factorization and various factorization identities. Varopoulos [Reference Varapoulos30, Reference Varapoulos31], Eichelsbacher and König [Reference Eichelsbacher and König11] and Denisov and Wachtel [Reference Denisov and Wachtel7, Reference Denisov and Wachtel8] have studied the setting of random walks in cones and have developed a new approach for obtaining exact asymptotics based on the construction of a harmonic function for a certain operator. This construction therefore avoids the use of the Wiener–Hopf factorization. Following this method, in the case of dependent random variables recent progress was made in [Reference Grama, Le Page and Peigné17, Reference Grama, Lauvergnat and Le Page15], where conditioned integral limit theorems for products of random matrices and for Markov chains satisfying spectral gap properties have been obtained. In [Reference Grama, Lauvergnat and Le Page16] a conditioned local limit theorem for a Markov chain with finite state space was considered.
As far as we know, conditioned integral and local limit theorems for Birkhoff sums have not yet been considered in the literature. In establishing these results we encountered two main difficulties.
The first is actually related with the statement of the conditioned limit theorems themselves. In the case of Markov chain, the statement of results requires the use of the corresponding harmonic function. In some cases, the subshift comes with an auxiliary Markov chain and the statement of the conditioned central limit theorem can be deduced from the Markov case. However, in general, to state the result for our dynamical system, we need a replacement for the harmonic function. Indeed, one of the major findings of the paper is that, in the case for the subshift of finite type $({\mathbb X}, T,\nu )$ with a general Hölder continuous observable f, a more general object, the harmonic measure $\mu ^f$ , has to be considered. The conditioned central limit theorem for the Birkhoff sum $t+S_nf$ is stated in terms of the harmonic measure $\mu ^f$ , whose use cannot be avoided and which constitutes an essential characteristic of the model. The construction of $\mu ^f$ is performed first for the sum $t+S_nf$ with an observable f depending only on the past coordinates, which in the reversed setting corresponds to studying a Markov chain. Then it is extended gradually to a function f depending on the whole set of coordinates using smoothing techniques and a vague convergence argument, see §3.
Once this construction is achieved we are able to adapt several statements from the Markov chain case, such as the conditioned central limit theorem, to the dynamical system setting. We were motivated by the previous developments in [Reference Grama, Lauvergnat and Le Page15, Reference Grama, Lauvergnat and Le Page16] for the Markov chains. To put it in a nutshell, we shall first establish the corresponding theorems for the Birkhoff sum $t+S_nf$ with an observable f depending only on the future coordinates, which corresponds to dealing with some Markov chain. Then we extend them to the general case of subshifts of finite type, using the technique similar to that developed for the proof of the existence of the harmonic measure.
The second difficulty is related to the proof of the corresponding conditioned local limit theorem. For proving the conditioned local limit theorem in the case of finite Markov chains [Reference Grama, Lauvergnat and Le Page16] it is necessary to consider the reversed walk, which in this particular case is again a Markov chain. For the subshift of finite type the situation is trickier, but can be handled using the reversed subshift. Once the harmonic measure $\mu ^f$ is constructed for any Hölder continuous observable f, this construction can be applied to the reversed subshift $({\mathbb X}, T^{-1}, \nu )$ with observable $-f \circ T^{-1}$ yielding the reverse harmonic measure $\check {\mu }^{-f}$ , which is necessary to state the conditioned local limit theorem. To prove the conditioned local limit theorem we are able to patch up the two conditioned integral limit theorems for the direct and reversed walks to establish a conditioned local limit theorem, where both measures $\mu ^{f}$ and $\check {\mu }^{-f}$ will show up. We use the techniques from [Reference Denisov and Wachtel7, Reference Grama and Xiao18] which deal with random walks with independent increments.
In a perspective, it is possible to apply the developed approach for studying conditioned local limit theorems for products of random matrices and more generally for Markov chains with values in general state spaces, in contrast to [Reference Grama, Lauvergnat and Le Page16] where a chain with finite state spaces has been considered. This will be the subject of a forthcoming paper.
2 Background and auxiliary statements
2.1 Subshift of finite type and Gibbs measure
We start by precisely introducing the subshift of finite type. Let $k \geqslant 2$ be an integer and $A=\{1, 2, \ldots , k\}$ . Let M be a transition matrix on A, that is, $M = (M (i,j))_{ i, j \in A }$ is a matrix with coefficients in $\{0, 1\}$ . We assume that the transition matrix M is aperiodic in the sense that there exists an integer $p \geqslant 1$ such that all the coefficients of the matrix $M^p$ are strictly positive. Consider the associated subshift of finite type
equipped with the shift map T defined by $(T x)_n = x_{n+1}$ for $x \in {\mathbb X}$ and $n \in {\mathbb Z}$ . The set $\{1, 2, \ldots , k\}$ is equipped with the discrete topology, so the space $A^{{\mathbb Z}}$ is compact with the corresponding Tychonov product topology. We equip ${\mathbb X}$ with the induced topology, which is also compact. For any $x = (x_n)_{n \in {\mathbb Z}} \in {\mathbb X}$ and $y = (y_n)_{n \in {\mathbb Z}} \in {\mathbb X}$ , define
Note that for any constant $\alpha \in (0,1)$ , the function $(x, y) \mapsto \alpha ^{\omega (x, y)}$ is a distance on ${\mathbb X}$ which induces the natural product topology.
The space of real-valued continuous functions $f: {\mathbb X} \to {\mathbb R}$ is denoted by $\mathcal C({\mathbb X})$ . For any function $f \in \mathcal C({\mathbb X})$ , we say that f is Hölder continuous on ${\mathbb X}$ if there exist constants $C>0$ and $\alpha \in (0,1)$ such that for all $x, y \in {\mathbb X}$ ,
For a fixed $\alpha \in (0,1)$ , denote by ${\mathscr B}_{\alpha }$ the space of all real-valued functions on ${\mathbb X}$ satisfying (2.1) for some constant C, equipped with the following norm
The function $d_{\alpha }: (x ,y) \mapsto \alpha ^{\omega (x,y)}$ is a distance on ${\mathbb X}$ , and ${\mathscr B}_{\alpha }$ is the space of Lipschitz continuous functions with respect to the distance $d_{\alpha }$ . Note that the notion of Lipschitz continuity depends on the index $\alpha $ , but the notion of Hölder continuity does not. It is clear that the set of all real-valued Hölder continuous functions on ${\mathbb X}$ can be written as ${\mathscr B} = \bigcup _{0 < \alpha < 1} {\mathscr B}_{\alpha }$ .
For any $f \in {\mathscr B}$ , we consider the Birkhoff sum process $(S_nf)_{n\geqslant 0}$ by setting $S_0f=0$ and
Let us denote by ${\mathbb X}^+ \subset A^{{\mathbb N}}$ the set
The set of continuous functions on ${\mathbb X}^+$ is denoted by $\mathcal {C}({\mathbb X}^+)$ . By abuse of notation, the one-sided shift map ${\mathbb X}^+ \mapsto {\mathbb X}^+$ will still be denoted by T.
The Ruelle operator $\mathcal L_f: \mathcal C({\mathbb X}^+) \to \mathcal C({\mathbb X}^+)$ related to $f \in \mathcal C({\mathbb X}^+)$ is defined as follows: for any $g \in \mathcal C({\mathbb X}^+)$ ,
One can easily see that $\mathcal L_{f}$ is a bounded linear operator on $\mathcal C({\mathbb X}^+)$ . From (2.3), by iteration, it follows that for any $n \geqslant 1$ ,
In addition, if $h \in \mathcal {C}({\mathbb X}^+)$ , we have the conjugacy relation
which tells us that the spectral properties of the transfer operator $\mathcal L_{f}$ only depend on the cohomology class of f. We say that a real-valued and Hölder continuous function $\psi $ on ${\mathbb X}^+$ is normalized if . By [Reference Parry and Pollicott24, Ch. 2, Theorem 2.2], there exist a real-valued Hölder continuous function h and a real number $\unicode{x3bb} $ such that $\mathcal L_{\psi } e^{h} = e^{\unicode{x3bb} + h}$ . From the conjugacy relation (2.4), this tells us that the function $\psi - h\circ T + h +\unicode{x3bb} $ is also normalized. Therefore, throughout the paper, we assume that $\psi $ is normalized. In this case, it is well known (e.g. [Reference Parry and Pollicott24]) that the adjoint operator $\mathcal L^*_{\psi }$ admits a unique invariant probability measure $\nu ^+$ . The measure $\nu ^+$ is called the Gibbs measure related to the potential $\psi $ . As $\psi $ is normalized, the measure $\nu ^+$ is T-invariant, that is, for any $f\in \mathcal C({\mathbb X}^+)$ ,
see [Reference Parry and Pollicott24, Ch. 2].
Note that $\nu ^+$ is also T-ergodic, meaning that any T-invariant Borel subset B of ${\mathbb X}^+$ has $\nu ^+$ measure $0$ or $1$ :
Thanks to the following lemma, the measure $\nu ^+$ allows to define a T-invariant measure on ${\mathbb X}$ .
Lemma 2.1. Let $\nu ^+$ be a Borel probability measure on ${\mathbb X}^+$ which is T-invariant. Then there exists a unique T-invariant Borel probability measure $\nu $ on ${\mathbb X}$ such that the image of $\nu $ under the natural projection map ${\mathbb X} \to {\mathbb X}^+$ is equal to $\nu ^+$ .
The proof of this lemma is just a consequence of Kolmogorov’s extension theorem. We actually give an explicit formula for the measure $\nu $ in the following.
2.2 Conditional measures on the past
For any $z\in {\mathbb X}^+ $ , we shall construct a measure $\nu _z^- $ , which is the conditional measure of $\nu $ with respect to the map $x\in {\mathbb X} \mapsto x_+ \in {\mathbb X}^+$ . To this end, for any $a \in A$ , let
where M is the transition matrix on the set A which was used to define the finite-type subshift ${\mathbb X} \subset A^{{\mathbb Z}}$ . For any $z \in {\mathbb X}^+$ , we set ${\mathbb X}^-_z = {\mathbb X}^-_{z_0}$ , where $z_0$ is the first coordinate of $z \in {\mathbb X}^+$ . We have the decomposition
The point z may be thought of as the future of the trajectory whereas the elements of ${\mathbb X}^-_z$ describe the pasts which are compatible with this future. Let us introduce some notation related to this decomposition. For any $z \in {\mathbb X}^+$ and $y \in {\mathbb X}^-_z$ , we denote $y\cdot z = (y, z) \in {\mathbb X}$ . For $z\in {\mathbb X}^+ $ and $k \geqslant 1$ , we set
For $(y_{-k}, \ldots , y_{-1}) \in A^k_z$ , we set $ y_{-k} \ldots y_{-1} \cdot z $ to be the element $w \in {\mathbb X}^+$ defined by
For $a \in A^k_z$ , let
be the associated cylinder of length k in ${\mathbb X}^-_z$ .
Recall that the two-sided shift map $T: {\mathbb X} \to {\mathbb X} $ and its inverse $T^{-1}$ are defined by $(T x)_n = x_{n +1}$ and $(T^{-1} x)_n = x_{n-1}$ for any $x \in {\mathbb X}$ and $n \in {\mathbb Z}$ . By abuse of notation, the one-sided forward shift map will be denoted by $T: {\mathbb X}^+ \mapsto {\mathbb X}^+ $ and is defined by $T(x) =(x_{1},x_{2},x_{3},\ldots ),$ for any $x=(x_{0},x_{1},x_{2},\ldots )\in {\mathbb X}^+$ . Let us define conditional measures on the past of trajectories. For $k \geqslant 0$ , define $\nu ^k_{z}$ as a function on cylinders of length k in ${\mathbb X}^-_z$ by the formula
for $a\in A_z^k$ . As , we have that for any $a \in A^k_z$ ,
By Kolmogorov’s extension theorem, from equation (2.8) it follows that there exists a unique Borel probability measure $\nu ^-_z$ on ${\mathbb X}^-_z$ such that for any $k \geqslant 0$ , $\nu ^k_z$ is the restriction of $\nu ^-_z$ to cylinders of length k.
We can now give an explicit formula for the measure $\nu $ in terms of the measures $\nu ^+$ and $\nu _z^-$ .
Lemma 2.2. Let $\varphi \in \mathcal {C({\mathbb X})}$ . Then we have
Proof. By Lemma 2.1, it suffices to prove that the measure $\nu $ on ${\mathbb X}$ defined by the above equation is T-invariant. This property is a direct consequence of the definition of the measures $\nu _z^-$ , $z \in {\mathbb X}^+$ , and of the fact that $\nu ^+$ is $\mathcal {L}_{\psi }$ -invariant.
We use the fact that the measures $\nu ^-_z$ and $\nu ^-_{z'}$ are equivalent.
Lemma 2.3. There exists a real-valued continuous function $\theta $ on the set
such that for any $z, z' \in {\mathbb X}^+$ and any continuous function $\varphi $ on ${\mathbb X}^-_z$ , one has
In addition, there exists a constant $c>0$ such that for any $(y, z, z') \in {\mathbb X}_3$ ,
Proof. Indeed, it suffices to set
2.3 General properties of exit times
From the following lemma it follows that the function $x \mapsto \tau _t^f(x)$ is finite $\nu $ -almost surely.
Lemma 2.4. Let $f \in {\mathscr B}$ with $\nu (f) = 0$ . Assume that f is not a coboundary. Then for $\nu $ -almost every $x \in {\mathbb X}$ , one has
Proof. Consider the Borel set
It is clear that the set A is T-invariant. Therefore, $\nu (A) = 0$ or $\nu (A) = 1$ . Assume that $\nu (A) = 1$ , then let us show that f is a coboundary. Indeed, for any $x \in A$ , we have that $h(x): = \liminf _{n \to \infty } S_n f(x)> -\infty $ . As $\nu (f) = 0$ , it is well known that $S_n f(x)$ admits finite limit points for $\nu $ -almost every $x \in {\mathbb X}$ , so that $h(x) < \infty $ . Now, by definition, we have $h(Tx) = h(x) - f(x)$ , hence f is a coboundary as a measurable function on ${\mathbb X}$ . Therefore, by [Reference Parry and Pollicott24, Proposition 6.2], we get that f is a coboundary as a Hölder continuous function on ${\mathbb X}$ .
For notational reasons, it is more convenient to study objects defined by the reverse shift $T^{-1}$ . Note that the two studies are equivalent.
Indeed, let us define the flip map $\iota : A^{{\mathbb Z}} \to A^{{\mathbb Z}}$ by the following relation: for any ${x = (\ldots , x_{-1}, x_0, x_{1}, \ldots )\in A^{{\mathbb Z}}}$ it holds $\iota (x) = (\ldots ,x_{1}, x_0, x_{-1},\ldots ) \in A^{{\mathbb Z}}$ , that is $(\iota x)_n = x_{-n}$ for $n \in {\mathbb Z}$ . The following lemma is classical (see [Reference Parry and Pollicott24, Ch. 2]).
Lemma 2.5. The set $\iota {\mathbb X}$ is a subshift of finite type and the measure $\iota _* \nu $ is a Gibbs measure on $\iota {\mathbb X}$ .
For $f \in {\mathscr B}$ , consider the reversed Birkhoff sum process $(\check S_n f)_{n \geqslant 1}$ which is defined as follows: for any $x\in {\mathbb X}$ ,
In the same way, denote by $ \check {\tau }_t^f(x)$ the first time when $t + \check S_n f(x)$ becomes negative: for any $x\in {\mathbb X}$ ,
Then the relation between the exit times $\tau _t^f$ and $\check \tau _t^{f \circ \iota }$ is given by
In the present paper we deal with the measure
which, by the previous discussion, is equivalent to studying the measure
In turn, Lemma 2.2 shows that in order to study (2.11), it suffices to investigate
for $z \in {\mathbb X}^+$ . We do it by using tools from the theory of Markov chains [Reference Grama, Lauvergnat and Le Page15]. In particular, we make use of the martingale approximation for the process $(\check S_n f)_{n \geqslant 1}$ .
2.4 Martingale approximation
Recall that ${\mathscr B} = \bigcup _{0 < \alpha < 1} {\mathscr B}_{\alpha }$ , where ${\mathscr B}_{\alpha }$ is the space of real-valued $\alpha $ -Hölder continuous functions on ${\mathbb X}$ endowed with the norm (2.2). In the same way, we denote by ${\mathscr B}^+_{\alpha }$ the space of real-valued $\alpha $ -Hölder continuous functions on ${\mathbb X}^+$ endowed with the norm
Let ${\mathscr B}^+ = \bigcup _{0 < \alpha < 1} {\mathscr B}_{\alpha }^+$ . Note that every Hölder continuous function f on ${\mathbb X}^+$ can be extended to a Hölder continuous function on ${\mathbb X}$ defined by
so we can identify ${\mathscr B}^+$ with a subspace of ${\mathscr B}$ .
Let $f\in {\mathscr B}$ . Define the cohomology class of f as the following set of Hölder continuous functions:
The following proposition tells us that there exists a natural choice in ${\mathscr C}(f)$ .
Proposition 2.6. Let $f\in {\mathscr B}$ be such that $\nu (f)=0$ . Then there exists a unique function $f_0 \in {\mathscr B}^+ $ such that $\mathcal L_{\psi } f_0 = 0$ and its extension on ${\mathbb X}$ belongs to ${\mathscr C} (f)$ .
Proof. First we prove the existence of $f_0$ . By Proposition 1.2 in [Reference Parry and Pollicott24], there exists a Hölder continuous function g on ${\mathbb X}^+$ , whose extension to ${\mathbb X}$ is cohomologous to f. As $\nu (f)=0$ , we have $\nu ^+ (g)=0$ . Now we choose $\alpha \in (0,1)$ close enough to $1$ so that $\mathcal L_{\psi }$ is bounded on ${\mathscr B}_{\alpha }$ and $g\in {\mathscr B}_{\alpha }$ . By the spectral gap property for the operator $\mathcal L_{\psi }$ (see Theorem 2.2 of [Reference Parry and Pollicott24]), there exists a Hölder continuous function $h \in {\mathscr B}_{\alpha }$ such that
As , it follows that
Hence, there exists a function $f_0: = g - h \circ T + h \in {\mathscr C}(f)$ satisfying $\mathcal L_{\psi } f_0 = 0$ .
Now we prove the uniqueness of $f_0$ . Suppose that there exist $f_0, f_0' \in {\mathscr C}(f)$ such that $\mathcal L_{\psi } f_0 = \mathcal L_{\psi } f_0' = 0$ . Then $f_0 - f_0'$ is a coboundary, namely, there exists $h_1 \in {\mathscr B}$ such that $f_0 - f_0' = h_1 \circ T - h_1$ . As $f_0$ and $f_0'$ depend only on the future coordinates, it is well known that $h_1$ depends only on the future coordinates. It follows that $\mathcal L_{\psi } (h_1 \circ T - h_1) = 0$ and, hence, $\mathcal L_{\psi } h_1 = h_1$ . This implies that $h_1$ is a constant and, therefore, $f_0' = f_0$ .
For any $z \in {\mathbb X}^+$ , we have defined a probability measure $\nu ^-_z$ on the set ${\mathbb X}^-_z \subset A^{- {\mathbb N}^*}$ of past sequences which are compatible with z. For $n \geqslant 1$ , we let $\mathscr F_n$ denote the $\sigma $ -algebra of subsets of $A^{- {\mathbb N}^*}$ generated by the coordinate maps $y \mapsto (y_{-1}, \ldots , y_{-n})$ . By convention, we also define $\mathscr F_0$ as the trivial $\sigma $ -algebra. We let $\mathscr F_n^z$ be the $\sigma $ -algebra induced on ${\mathbb X}^-_z$ . The following proposition is a classical result from [Reference Parry and Pollicott24]:
Proposition 2.7. Let $f_0\in \mathcal {C} ({\mathbb X}^+)$ . Then $\mathcal L_{\psi } f_0 = 0$ if and only if for any $z\in {\mathbb X}^+$ , the sequence of random variables
is a martingale on ${\mathbb X}^-_z$ equipped with the probability measure $\nu ^-_z$ with respect to the filtration $(\mathscr F_n^z)_{n \geqslant 0}$ .
Proof. Denote by $g_n^z: {\mathbb X}^-_z \to {\mathbb R}$ the function $y \mapsto \check S_n f_0 ( y \cdot z )$ . Then for $y \in {\mathbb X}^-_z$ and ${n \geqslant 1}$ , we have by the definition of the measure $\nu ^-_z$ ,
From this identity, the assertion follows.
The following result shows that the difference $\check S_n f - \check S_n g$ is bounded, for f and g in the same cohomology class.
Lemma 2.8. Let $f\in {\mathscr B}$ and $g\in {\mathscr C}(f) $ . Let $h\in {\mathscr B}$ be such that $f-g= h\circ T-h$ . Then, for any $x\in {\mathbb X}$ and any $n \geqslant 1$ , we have
Proof. Indeed, we have $S_n f - S_n g = h \circ T^{n} - h.$ As $\check S_n f = (S_n f) \circ T^{-n}$ , we obtain $\check S_n f - \check S_n g = h - h \circ T^{-n}$ , which proves the assertion.
2.5 The Hölder continuity and approximation
We establish several technical results which will be used in the proofs of the main results of the paper. In particular, they allow us to prove that several convergences hold uniformly in $z \in {\mathbb X}^+$ .
Lemma 2.9. For any $g \in {\mathscr B}$ , there exist constants $\alpha \in (0,1)$ and $c_0>0$ such that for any $n \geqslant 1$ , $z, z' \in {\mathbb X}^+$ with $z_0 = z_0'$ and $y \in {\mathbb X}^-_z (= {\mathbb X}^-_{z'})$ , one has
In particular, for any $g \in {\mathscr B}$ , there exists a constant $c_0>0$ such that for any $n \geqslant 1$ , $z, z' \in {\mathbb X}^+$ with $z_0 = z_0'$ and $y \in {\mathbb X}^-_z (= {\mathbb X}^-_{z'})$ , it holds
Proof. As $g \in {\mathscr B}$ , there exists a constant $L_{g}$ such that for any $x, x' \in {\mathbb X}$ ,
where $0< \alpha <1$ . Hence, for any $z, z' \in {\mathbb X}^+$ with $z_0 = z_0'$ and $y \in {\mathbb X}^-_z$ , and $n \geqslant 1$ , one has
The desired result follows.
Corollary 2.10. For any $g \in {\mathscr B}$ , there exist constants $\alpha \in (0,1)$ and $c_0>0$ such that for any $n \geqslant 1$ , $z, z' \in {\mathbb X}^+$ with $z_0 = z_0'$ and $y \in {\mathbb X}^-_z (= {\mathbb X}^-_{z'})$ , we have
Proof. By Lemma 2.9, there exist constants $c_0>0$ and $\alpha \in (0,1)$ such that for any $n \geqslant j \geqslant 1$ ,
Taking the minimum over $1 \leqslant j \leqslant n$ on the right-hand side, we get
In the same way, again by Lemma 2.9, there exist constants $c_0>0$ and $\alpha \in (0,1)$ such that for any $n \geqslant j \geqslant 1$ ,
Taking the minimum over $1 \leqslant j \leqslant n$ on the left-hand side, we get
Combining (2.16) and (2.17), we conclude the proof of (2.15).
We also need the following technical lemma that allows us to approximate the function $g \in {\mathscr B}$ by a function $x \mapsto g_m(x)$ on ${\mathbb X}$ which only depends on the coordinates $\{x_k\}_{k \geqslant -m}$ .
Lemma 2.11. Let $g \in {\mathscr B}$ . Then there exist constants $\alpha \in (0,1)$ , $c_1>0$ and a sequence of Hölder continuous functions $(g_m)_{m \geqslant 0}$ on ${\mathbb X}$ which only depend on the coordinates $\{x_k\}_{k \geqslant -m}$ such that $\mathcal {L}_{\psi } g_0 = 0$ and for any $m \geqslant 0$ ,
Proof. By Proposition 2.6, there exist $g_0 \in {\mathscr B}^+$ and $h \in {\mathscr B}$ with $\mathcal {L}_{\psi } g_0 = 0$ and
As $h \in {\mathscr B}$ , there is $\alpha \in (0,1)$ such that $h \in {\mathscr B}_{\alpha }$ . Then, for any $m\geqslant 0$ , there exists a Hölder continuous function $h_m$ on ${\mathbb X}$ which only depends on the coordinates $\{x_k\}_{k \geqslant -m}$ such that
where $c_1>0$ is a fixed constant not depending on m and, by convention, $h_0 = 0$ . We define for any $m \geqslant 0$ ,
2.6 Duality
The next duality property is crucial in the proof of the main results.
Lemma 2.12. Let $g \in {\mathscr B}$ . For any $n\geqslant 1$ and any non-negative measurable function $F: {\mathbb X} \times {\mathbb R} \times {\mathbb X} \times {\mathbb R} \to {\mathbb R}$ , we have
Proof. By a change of variable $t = u - \check S_n g(x)$ , it follows that
As the measure $\nu $ is $T^{-1}$ -invariant, we obtain
which ends the proof of the lemma.
3 Harmonicity for dynamical system
3.1 Existence of the harmonic function
The aim of this section is to prove the existence of a function $V^{f}$ on the state space ${\mathbb R}$ which we call the harmonic function of f by analogy with the theory developed for Markov chains in [Reference Grama, Lauvergnat and Le Page15]. Our main result is the following theorem.
Theorem 3.1. Let f be a Hölder continuous function on ${\mathbb X}$ such that f is not a coboundary and $\nu (f) = 0$ . Then there exists a unique non-decreasing and right continuous function $V^{f}: {\mathbb R} \to {\mathbb R}_+$ such that for any continuous compactly supported function $\varphi $ on ${\mathbb R}$ ,
In addition, there exists a constant $c>0$ such that for any $t\in {\mathbb R}$ ,
Note that the bound (3.2) implies that $V^f(t)/t \to 1$ as $t \to \infty $ .
The proof of Theorem 3.1 is given at the end of this section. At this point, we start by giving an explicit formula for the harmonic function in the case where the observable only depends on future coordinates. Let $g \in {\mathscr B}^+$ with $\nu (g) = 0$ and assume that g is not a coboundary. Let $g_0$ be the unique element of ${\mathscr B}^+$ such that $\mathcal {L}_{\psi } g_0 = 0$ and $g_0$ is cohomologous to g, as in Proposition 2.6. For $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ , we set
This integral makes sense. Indeed, first, by Lemma 3.3, the stopping time $y \mapsto \check {\tau }_{t}^{g} (y \cdot z)$ is finite $\nu ^-_z$ -almost everywhere. Second, the Birkhoff sum $t + \check {S}_{\check {\tau }_{t}^{g} (y \cdot z)} g (y \cdot z)$ takes values in the interval $[ -\|g\|_{\infty }, 0]$ when t is non-negative, and in the interval $[t-\|g\|_{\infty }, 0]$ when t is negative. Third, by Lemma 2.8, the difference of the Birkhoff sums for g and $g_0$ is uniformly bounded.
The function $\check V^{g}(z, \cdot )$ plays a crucial role in proving conditioned limit theorems for products of random matrices and more generally for Markov chains, see [Reference Grama, Lauvergnat and Le Page15, Reference Grama, Le Page and Peigné17]. From the results of [Reference Grama, Lauvergnat and Le Page15] it follows that $\check V^{g}(z, \cdot )$ has the following harmonicity property.
Lemma 3.2. Let g be in ${\mathscr B}^+$ such that $\nu ^+(g) = 0$ and g is not a coboundary. Then for any $(z,t) \in {\mathbb X}^+ \times {\mathbb R}$ , we have
The proof of the existence of the harmonic function $\check V^g$ given in [Reference Grama, Lauvergnat and Le Page15] is rather difficult. In the case of the subshift of finite type (because the jumps are bounded) it is possible to give a much shorter direct proof, which is not included because of the space limitations.
We extend the definition of $\check V^{g}(z, \cdot )$ to the case of any function $g \in {\mathscr B}$ , that is, the case of a function g that depends on both the past and the future coordinates. We use the following technical assertion.
Lemma 3.3. Let $g \in {\mathscr B}$ such that $\nu (g) = 0$ and g is not a coboundary with respect to T. Then, for any $t \in {\mathbb R}$ , it holds uniformly in $z \in {\mathbb X}^+$ that
Proof. Let $c_0>0$ be as in Lemma 2.9. By Lemma 2.4 and Fubini’s theorem, for any $a \in A$ , we can find $z' \in {\mathbb X}^+$ such that $z^{\prime }_0 = a$ and the function $y \mapsto \check \tau _{t+c_0}^g(y \cdot z')$ on ${\mathbb X}^-_{z'}$ is finite $\nu ^-_{z'}$ -almost everywhere. Then for any $z \in {\mathbb X}^+$ with $z_0 = a$ , we have
From Lemma 2.3, we get that for some constant $c>0$ ,
Thus, the lemma follows from the fact that $\nu ^-_{z'}( y \in {\mathbb X}^-_{z'}: \check {\tau }_{t+c_0}^g(y \cdot z')>n )$ converges to $0$ as $n \to \infty $ .
Now we give an alternative definition of the function $\check V^{g}(z, \cdot )$ for $g \in {\mathscr B}^+$ , where the key point is that in this case, the function $y \mapsto \check {\tau }_t^{g} (y \cdot z)$ is a stopping time with respect to the filtration $\{ \mathscr {F}_k^z \}_{k \geqslant 0}$ .
Lemma 3.4. Let $g \in {\mathscr B}^+$ with $\nu (g) = 0$ and assume that g is not a coboundary. Let $g_0$ be the unique element of ${\mathscr B}^+$ such that $\mathcal {L}_{\psi } (g_0) = 0$ and $g_0$ is cohomologous to g. Then, for any $t \in {\mathbb R}$ , we have, uniformly in $z \in {\mathbb X}^+$ ,
In addition, there is a constant $c>0$ such that, for any $z \in {\mathbb X}^+$ , $t \in {\mathbb R}$ and $n\geqslant 1$ ,
for any $z \in {\mathbb X}^+$ and $t \in {\mathbb R}_+$ ,
and for any $z \in {\mathbb X}^+$ and $t < -c$ , it holds that $\check V^{g_0}(z, t) = 0$ .
Moreover, for any $z \in {\mathbb X}^+$ , the function $\check V^{g}(z, \cdot )$ is non-decreasing on $\mathbb {R}$ .
Proof. As g is cohomologous to $g_0$ , by Lemma 2.8, all Birkhoff sums $\check S_{n} g (y \cdot z)$ stay at bounded distance from the Birkhoff sums $\check S_{n} g_0 (y \cdot z)$ . Therefore, one can deal with $\check S_{n} g_0 (y \cdot z)$ instead of $\check S_{n} g (y \cdot z)$ . By the optional stopping theorem, for any $n\geqslant 1$ ,
The bound (3.6) follows because $t+ \check S_{\check {\tau }_t^{g} (y \cdot z)} g_0 (y \cdot z)$ is bounded from below, because ${t+ \check S_{\check {\tau }_t^{g} (y \cdot z)-1} g (y \cdot z) \geqslant 0}$ and g is bounded.
Note that, as $t+ \check S_{\check {\tau }_t^{g} (y \cdot z)} g (y \cdot z) < 0$ , the quantity $t+ \check S_{\check {\tau }_t^{g} (y \cdot z)} g_0 (y \cdot z)$ is also bounded from above. Therefore, by Lemma 3.3, we obtain (3.5) uniformly in $z \in {\mathbb X}^+$ .
Still because the function $y \mapsto t + \check S_{\check {\tau }_t^{g} (y \cdot z)} g_0 (y \cdot z)$ is uniformly bounded, we get $\check V^{g}(z, t) \in [t-c, t + c ]$ , for some constant $c>0$ . In addition, if $t < - \|g\|_{\infty }$ , we get $\check {\tau }_{t}^{g} (y \cdot z) = 1$ everywhere for all $z \in {\mathbb X}^+$ ; thus, by (2.7) and (2.3) we have
It remains to prove the monotonicity of $t\mapsto \check V^{g}(z, t) $ . As $\check {\tau }_{t_1}^{g} \leqslant \check {\tau }_{t_2}^{g}$ for any $t_1 \leqslant t_2$ , and $t_2 + \check S_n g \geqslant 0$ on the set $\{\check {\tau }_{t_2}^{g}>n\}$ , it follows that
Letting $n\to \infty $ yields that the function $\check V^{g}(z, \cdot )$ is non-decreasing on $\mathbb {R}$ .
By using Lemma 3.4, we can now give a definition of $\check {V}^g$ for a function g only depending on finitely many negative coordinates.
Lemma 3.5. Let $g \in {\mathscr B}$ such that $\nu (g) = 0$ and g is not a coboundary. Assume that g only depends on m negative coordinates for some $m \geqslant 0$ , in other words, that the function $h=g\circ t^m$ belongs to ${\mathscr B}^+$ . Then, for any $t \in {\mathbb R}$ , we have uniformly in $z \in {\mathbb X}^+$ ,
Let g and h be as in Lemma 3.5. We set for $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ ,
Lemma 3.5 implies that this notation is coherent with that introduced in (3.3).
Proof of Lemma 3.5.
By conditioning over the m first coordinates of y, we get, for $n \geqslant 0$ ,
where we have used the relations $(y \cdot a) \cdot z = T^m (y \cdot (a \cdot z))$ and $\check {\tau }_t^{h \circ T^{-m}} = \check {\tau }_t^{h} \circ T^{-m}$ . The conclusion now follows from Lemma 3.4 and the definition of the transfer operator $\mathcal L_{\psi }^m.$
We prove that the convergence in Lemma 3.5 holds in a weak sense for every function $g \in {\mathscr B}$ . The key step to prove Theorem 3.1 is the following technical lemma which shows that the convergence of Lemma 3.5 holds for all functions $g \in {\mathscr B}$ in a weak sense.
Lemma 3.6. Assume that $g\in {\mathscr B}$ is not a coboundary with respect to T and $\nu (g)=0$ . Then, for any continuous compactly supported function $\varphi $ on ${\mathbb R}$ , uniformly in $z \in {\mathbb X}^+$ , the following limit exists and is finite:
Proof. Assume that $g\in {\mathscr B}$ . Let $(g_m)_{m \geqslant 0}$ , $c_1>0$ and $\alpha \in (0,1)$ be as in Lemma 2.11. Set
and
By (2.18), we have the inclusions
which imply that
In the same way, we have
where the last bound follows from (3.6).
By Lemma 3.5, for fixed $m \geqslant 0$ , as $n \to \infty $ , the function $W_{n,m} (z, t)$ converges to $\check {V}^{g_m} (z, t)$ , uniformly in $z \in {\mathbb X}^+$ . From (3.8) we get
Now we have
Using (3.9) and Lemma 3.5, we have that $\check {V}^{g_m} (z, t) \leqslant c_2 + \max \{t, 0\}$ . As $\varphi $ is continuous on ${\mathbb R}$ with compact support, by the Lebesgue-dominated convergence theorem, we get that uniformly in $z \in {\mathbb X}^+$ ,
This tells us that $\int _{{\mathbb R}} \varphi (t) W_n(z,t) \,dt$ has a uniform limit as $n \to \infty $ .
We use the previous lemma to build a function $\check V^g(z,t)$ . The existence of this function will be deduced from the following elementary fact from the theory of distributions.
Lemma 3.7. Let $(V_n)_{n \geqslant 1}$ be a sequence of non-decreasing functions on ${\mathbb R}$ . Assume that for every continuous compactly supported function $\varphi $ on ${\mathbb R}$ , the sequence $\int _{{\mathbb R}} V_n(t) \varphi (t) \,dt$ admits a finite limit. Then there exists a unique right continuous and non-decreasing function V on ${\mathbb R}$ such that for any continuous compactly supported function $\varphi $ , we have
Now we construct the function $\check V^g(z,t)$ for any $g \in {\mathscr B}$ .
Lemma 3.8. Assume that $g\in {\mathscr B}$ is not a coboundary with respect to T and $\nu (g)=0$ . Then, for any $z \in {\mathbb X}^+$ , there exists a unique non-decreasing and right continuous function $\check V^{g}(z, \cdot )$ on ${\mathbb R}$ such that the following hold.
-
(1) For any continuous compactly supported function $\varphi $ on ${\mathbb R}$ , uniformly in $z \in {\mathbb X}^+$ ,
(3.10) -
(2) For any continuous compactly supported function $\varphi $ on ${\mathbb R}$ , the mapping ${z \mapsto \int _{{\mathbb R}} \varphi (t) \check V^{g}(z, t) \,dt}$ is continuous on ${\mathbb X}^+$ .
-
(3) There exists a constant $c>0$ such that for any $z \in {\mathbb X}^+$ and $t \in {\mathbb R}_+$ ,
(3.11) $$ \begin{align} t-c \leqslant \check V^{g}(z, t) \leqslant t + c. \end{align} $$In addition, for any $z \in {\mathbb X}^+$ and $t \leqslant -c$ , we have $\check V^{g}(z, t) = 0$ .
By Lemma 3.4, in the case $g \in {\mathscr B}^+$ , the notation $\check V^{g}(z, \cdot )$ is coherent with that in (3.3).
Proof of Lemma 3.8.
Fix $z \in {\mathbb X}^+$ . By Lemmas 3.3 and 3.6, the following limit exists: for any continuous compactly supported function $\varphi $ on ${\mathbb R}$ ,
For $t \in {\mathbb R}$ , set
Then the function $\check V_n^g (z, \cdot )$ is non-decreasing on ${\mathbb R}$ . By Lemma 3.7, there exists a unique non-decreasing and right continuous function $\check V^g (z, \cdot )$ on ${\mathbb R}$ such that for any continuous function $\varphi $ on ${\mathbb R}$ with compact support,
Note that for $t < - \|g\|_{\infty }$ , we have $\check {\tau }^{g}_t = 1$ everywhere. Hence, $\check V^{g}(z, t) = 0$ for ${t \leqslant - c}$ .
We now prove (3.11). By Proposition 2.6, there exists $g_0 \in {\mathscr B}^+$ such that $\mathcal {L}_{\psi } (g_0) = 0$ and g is cohomologous to $g_0$ . By Lemma 2.11, we can choose a constant $c>0$ large enough such that for any $n \geqslant 1$ , it holds that $\| \check S_n g - \check S_n g_0 \|_{\infty } \leqslant c$ . By Lemmas 3.3 and 3.6, we have, for any continuous non-negative function $\varphi $ on ${\mathbb R}$ with compact support,
Note that from Lemma 3.3, we have $\nu ^-_z( y \in {\mathbb X}^-_z: \check {\tau }_t^g(y \cdot z)>n) \to 0$ as $n \to \infty $ . As we have the following inclusion: for any $t \in {\mathbb R}$ ,
and as $t + c + \check S_{n} g_0 \geqslant 0$ on the set $\{ \check {\tau }_{t+c}^{g_0}>n \}$ , we obtain
As this holds for any continuous non-negative test function $\varphi $ on ${\mathbb R}$ , we obtain
This, together with Lemma 3.7, concludes the proof of (3.11).
We now want to prove the continuity in $z \in {\mathbb X}^+$ of the function $z \mapsto \int _{{\mathbb R}} \varphi (t) \check V^g (z,t) \,dt$ . To this aim, we establish a uniform bound for the quantity $\check V_n^g (z,t)$ defined in (3.13). Indeed, as usual, we have $\check V_n^g (z,t) \leqslant \check V_n^{g_0} (z,t+c)$ . Now the optional stopping theorem gives
From (3.14) we get
It remains to prove that for any continuous compactly supported function $\varphi $ on ${\mathbb R}$ , the mapping $z \mapsto \int _{{\mathbb R}} \varphi (t) \check V^{g}(z, t) \,dt$ is continuous on ${\mathbb X}^+$ . It suffices to prove that for any $n \geqslant 1$ , the mapping $z \mapsto \int _{{\mathbb R}} \varphi (t) \check V_n^{g}(z, t) \,dt$ is continuous on ${\mathbb X}^+$ .
A priori, for fixed $t \in {\mathbb R}$ , the function $z \mapsto \check V_n^{g}(z, t)$ is not continuous. Nevertheless, we claim that it satisfies the following weak continuity property: for $\varepsilon>0$ , there exists $k \in {\mathbb N}$ such that for any $z, z' \in {\mathbb X}^+$ with $w(z, z') \geqslant k$ we have
Indeed, this follows from the inequality (2.14) in Lemma 2.9. This, together with the bound (3.15) and the uniform continuity of the function $\varphi $ , implies that the mapping $z \mapsto \int _{{\mathbb R}} \varphi (t) \check V_n^{g}(z, t) \,dt$ is continuous on ${\mathbb X}^+$ .
The previous statements can be summarized as follows.
Theorem 3.9. Let g be a Hölder continuous function on ${\mathbb X}$ such that $\nu (g) = 0$ and g is not a coboundary. Then there exists a unique non-decreasing and right continuous function $\check V^{g}: {\mathbb R} \to {\mathbb R}_+$ with the following properties.
-
(1) For any continuous compactly supported function $\varphi $ on ${\mathbb R}$ ,
(3.16) -
(2) There exists a constant $c>0$ such that for any $t\in {\mathbb R}$ it holds
(3.17) $$ \begin{align} \max \{t - c, 0\} \leqslant \check V^g(t) \leqslant \max \{t, 0 \} + c. \end{align} $$
3.2 Properties of the harmonic function
The goal of this section is to give some additional properties of the harmonic function $\check V^{g}$ which will be necessary for the proof of Theorem 1.3. We start with a continuity result on the cohomology class of the function g.
Lemma 3.10. Let $g \in {\mathscr B}$ with $\nu (g) = 0$ . Assume that g is not a coboundary. Let $\alpha \in (0,1)$ and $(h_n)_{n \geqslant 0}$ be a sequence of element of ${\mathscr B}_{\alpha }$ that converges to $0$ with respect to the Hölder norm $\|\cdot \|_{\alpha }$ . For $n \geqslant 0$ , set $g_n = g + h_n \circ T - h_n$ . Then, there exists a constant $c>0$ such that for any $n \geqslant 0$ , $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ , one has
Moreover, for any continuous compactly supported function $\varphi $ on ${\mathbb R}$ , we have, uniformly in $z \in {\mathbb X}^+$ ,
Proof. The bound (3.18) follows from (3.11) and the relation $g_n = g + h_n \circ T - h_n$ . The construction of the function $\check V^{g}$ in (3.19) can be performed in the same way as in Lemmas 3.6 and 3.8.
We can also describe how the function $\check V^{g}$ behaves when the function g is shifted by the dynamics.
Lemma 3.11. Let $g \in {\mathscr B}$ with $\nu (g) = 0$ . Assume that g is not a coboundary. Then, for any $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ , we have
Proof. By Lemma 3.8, for any continuous compactly supported function $\varphi $ on ${\mathbb R}$ , we have
By conditioning on the coordinate $y_{-1}$ , we get
Again by Lemma 3.8, we obtain
as desired.
3.3 The harmonic measure and the proof of Theorem 1.1
In the case when g depends only on the future ( $g \in {\mathscr B}^+$ ), the function $\check V^g$ satisfies the harmonicity equation (3.4). In general, when g depends also on the past, this property may not hold. It turns out that equation (3.4) can be reinterpreted as a kind of invariance property of a certain Radon measure, which we introduce at the end of this section. Indeed, we have:
Lemma 3.12. Let g be in ${\mathscr B}^+$ and let V be a locally integrable non-negative function on ${\mathbb X}^+ \times {\mathbb R}$ . Then the following are equivalent.
-
(1) For $\nu ^+ \otimes \,dt$ almost every $(z, t)$ in ${\mathbb X}^+ \times {\mathbb R}$ , we have
-
(2) For any continuous compactly supported function $\varphi $ on ${\mathbb X}^+ \times {\mathbb R}$ , we have
(3.20) $$ \begin{align} \int_{{\mathbb X}^+ \times {\mathbb R}} \varphi(z, t) V(z, t) \nu^+(dz) \,dt = \int_{{\mathbb X}^+} \int_{0}^{\infty} \varphi(Tz, t - g(z)) V(z, t) \nu^+(dz) \,dt. \end{align} $$
Proof. The proof is a direct computation. Indeed, for any continuous compactly supported function $\varphi $ on ${\mathbb X}^+ \times {\mathbb R}$ , by a change of variable, the right-hand side of (3.20) can be written as
As $\nu ^+$ is $\mathcal L_{\psi }$ invariant, by using (2.5), we get for $t \in {\mathbb R}$ ,
This proves the lemma.
We now show that the functions $\check V^{g}$ and $V^{g}$ can be seen as the densities with respect to the Lebesgue measure on ${\mathbb R}$ of the projections on ${\mathbb R}$ of certain natural Radon measures $\check \mu ^g$ and $\mu ^{g}$ on ${\mathbb X} \times {\mathbb R}$ , which satisfy an invariance property similar to (3.20). Those measures will play a key role in the statement of the conditioned local limit theorem. The purpose of this subsection is to build them. This construction will follow the same lines as that of the harmonic functions. We first use Markov chain arguments to define these objects when $g \in {\mathscr B}^+$ and then use approximation arguments to extend the definition to the general case.
We first assume that g is in ${\mathscr B}^+$ . In that case, for $(z, t) \in {\mathbb X}^+ \times {\mathbb R}$ with $\check V^{g} (z,t)>0$ , let us introduce a Borel probability measure $\check \mu ^{g, -}_{z, t}$ on ${\mathbb X}^-_z$ . To do this, for $n \geqslant 1$ , let $A^n_z$ be as in the definition (2.7). For $a \in A^n_z$ , let us write $a \cdot z$ for the element ${\mathbb X}^+$ whose n first coordinates are $a_{-n}, \ldots , a_{-1}$ and whose kth coordinate is $z_{k-n}$ for $k \geqslant n$ .
Lemma 3.13. Let g be in ${\mathscr B}^+$ such that $\nu ^+(g) = 0$ and g is not cohomologous to $0$ . Let $(z, t)$ be in ${\mathbb X}^+ \times {\mathbb R}$ with $\check V^{g} (z,t)>0$ . Then, there exists a unique Borel probability measure $\check \mu ^{g, -}_{z, t}$ on ${\mathbb X}^-_z$ such that for any $n \geqslant 0$ and any $a \in A^n_z$ we have
as soon as $t + S_k g (T^k(a \cdot z)) \geqslant 0$ for all $1 \leqslant k \leqslant n$ .
Proof. The proof is a translation of the general construction of the Markov measures on the set of trajectories of a Markov chain.
Recall that, for $a \in A^n_z$ , we denoted by $\mathbb C_{a, z}$ (see (2.6)) the associated cylinder of length n in ${\mathbb X}^-_z$ . For $n \geqslant 0$ , define $\check \mu ^{g, n}_{z, t}$ as a function on cylinders of length n in ${\mathbb X}^-_z$ by the formula
if $t + S_k g (T^k(a \cdot z)) \geqslant 0$ for all $1 \leqslant k \leqslant n$ ; if not, we set $\check \mu ^{g, n}_{z, t} (\mathbb C_{a, z}) = 0$ , (compare with (2.7)). We claim that for any $a \in A^n_z$ , we have
(compare with (2.8)). Indeed, this follows from the harmonicity property of the function $\check V^g$ established in Lemma 3.2. By Kolmogorov’s extension theorem, equation (3.22) implies that there exists a unique Borel probability measure $\check \mu ^{g, -}_{z, t}$ on ${\mathbb X}^-_z$ such that for any $n \geqslant 0$ , $\check \mu ^{g, n}_{z, t}$ is the restriction of $\check \mu ^{g, -}_{z, t}$ to cylinders of length n. The lemma follows.
In the same way as for the function $\check V^{g}$ , we can give an alternative definition of the measures $\check \mu ^{g, -}_{z, t}$ , which relies on a convergence property.
Lemma 3.14. Let $g \in {\mathscr B}^+$ with $\nu (g) = 0$ and assume that g is not a coboundary. Let $(z, t)$ be in ${\mathbb X}^+ \times {\mathbb R}$ and $\varphi $ be a continuous function on ${\mathbb X}^-_z$ . Then, we have
Proof. By Lemma 3.3, the limit in equation (3.23) is the same as the limit of
The latter quantity is non-negative whenever $\varphi $ is non-negative. In addition, if
, the convergence follows from Lemma 3.4. Therefore, it suffices to check the convergence when $\varphi $ is the indicator function of a cylinder set. Thus, let $m \geqslant 0$ be an integer. Pick $a \in A^m_z$ and let $\mathbb C_{a,z}$ be the associated cylinder in ${\mathbb X}^-_z$ . If $S_k g (T^k(a \cdot z)) < 0$ for some $1 \leqslant k \leqslant m$ , we have for $n \geqslant m$ ,
If not, we have for $n \geqslant m$ ,
By Lemma 3.4, as $n \to \infty $ , this converges to
which, by the definition of $\check \mu ^{g, -}_{z, t}$ in Lemma 3.13, is equal to $\check \mu ^{g, -}_{z, t} (\mathbb C_{a,z}) \check V^g (z,t)$ .
Using Lemma 3.14, we can now give a definition of $\check \mu ^{g, -}_{z, t}$ for a function g only depending on finitely many negative coordinates.
Lemma 3.15. Let $g \in {\mathscr B}$ such that $\nu (g) = 0$ and g is not a coboundary. Assume that g only depends on m negative coordinates for some $m \geqslant 0$ . In other words, the function $h = g \circ T^{m} \in {\mathscr B}^+$ . Let $(z, t)$ be in ${\mathbb X}^+ \times {\mathbb R}$ and $\varphi $ be a continuous function on ${\mathbb X}^-_z$ . For $a \in A^m_z$ , set $\varphi _a$ to be the function $y \mapsto \varphi (y \cdot a)$ on ${\mathbb X}^-_{a \cdot z}$ . Then, we have
Before proving this lemma, we recall some useful facts. Let g and h be as in Lemma 3.15. For $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ ,
If $\check {V}^g (z,t)>0$ and $\varphi $ is a continuous function on ${\mathbb X}^-_z$ , we set
Lemma 3.15 implies that the notation (3.24) is coherent with that introduced in Lemma 3.13.
Proof of Lemma 3.15.
As in the proof of Lemma 3.5, by conditioning over the m first coordinates of y, we get for $n \geqslant 0$ ,
where we have used the relations $(y \cdot a) \cdot z = T^m (y \cdot (a \cdot z))$ and $\check {\tau }_t^{h \circ T^{-m}} = \check {\tau }_t^{h} \circ T^{-m}$ . The conclusion now follows from Lemma 3.14.
Now we prove that the convergence in Lemma 3.15 holds in a weak sense for every function $g \in {\mathscr B}$ .
Lemma 3.16. Assume that $g\in {\mathscr B}$ is not a coboundary with respect to T and $\nu (g)=0$ . Then, for any $z \in {\mathbb X}^+$ , for any continuous compactly supported function $\varphi $ on ${\mathbb X}^-_z \times {\mathbb R}$ , the following limit exists and is finite:
Proof. First let us assume that $\varphi $ is of the form $(y, t) \mapsto \varphi _1(y) \varphi _2(t)$ , where $\varphi _1$ and $\varphi _2$ are non-negative continuous functions on ${\mathbb X}^-_z$ and ${\mathbb R}$ , and $\varphi _2$ is compactly supported. In that case, let $(g_m)_{m \geqslant 0}$ , $c_1>0$ and $\alpha \in (0,1)$ be as in Lemma 2.11. Set
and
By (2.18), we have the inclusions
which imply that
By Lemma 3.15, for fixed $m \geqslant 0$ , as $n \to \infty $ , the function $W_{n,m} (z, t)$ converges to $\check \mu ^{g_m, -}_{z, t} (\varphi _1) \check {V}^{g_m} (z, t)$ . From (3.25) we get
Now we have
Using (3.9) and Lemma 3.5, we have that $\check {V}^{g_m} (z, t) \leqslant c_2 + \max \{t, 0\}$ . As $\varphi _2$ is continuous on ${\mathbb R}$ with compact support, by the Lebesgue-dominated convergence theorem, we get that the left-hand side of (3.26) converges to $0$ as $m \to \infty $ . This tells us that $\int _{{\mathbb R}} \varphi _2(t) W_n(z,t) \,dt$ has a limit as $n \to \infty $ . In other words, the lemma holds for the function $\varphi (y,t) = \varphi _1(y) \varphi _2(t)$ . This is also true when $\varphi _1$ and $\varphi _2$ are not necessarily non-negative.
The general case follows from a standard but tedious approximation argument. Indeed, we can find a continuous compactly supported function $\theta $ on ${\mathbb R}$ with support K such that for any $\varepsilon>0$ , there exist an integer $p \geqslant 0$ and continuous functions $\varphi _{i, 1}$ on ${\mathbb X}^-_z$ and continuous compactly supported functions $\varphi _{i, 2}$ on ${\mathbb R}$ with support included in K, $1 \leqslant i \leqslant p$ , with
where $\varphi _{\varepsilon } (y,t) = \sum _{i = 1}^p \varphi _{i, 1}(y) \varphi _{i, 2}(t)$ . We set $t_0 = \sup _{t \in K} |t|$ . By Lemma 3.3, we need to show that
has a limit as $n \to \infty $ . By the first case, we know that
has a limit $U_{\varepsilon }$ as $n \to \infty $ . In addition, by Lemma 3.6, we get that
converges to $\int _{{\mathbb R}} \check V^g (z,t) \theta (t) \,dt$ . By (3.27), we have
which gives
Hence, the proof of Lemma 3.16 is complete.
Now we use the previous lemma to build a Radon measure $\check {\mu }_{z}^{g, -}$ on ${\mathbb X}_z^- \times {\mathbb R}$ for any $g \in {\mathscr B}$ .
Lemma 3.17. Assume that $g\in {\mathscr B}$ is not a coboundary with respect to T and $\nu (g)=0$ . Then, for any $z \in {\mathbb X}^+$ , there exists a unique Radon measure $\check {\mu }_{z}^{g, -}$ on ${\mathbb X}_z^- \times {\mathbb R}$ such that for any continuous compactly supported function $\varphi $ on ${\mathbb X}_z^- \times {\mathbb R}$ ,
In addition, the marginal measure of $\check {\mu }_{z}^{g, -}$ on ${\mathbb R}$ under the natural projection map is the absolutely continuous measure $\check V^g(z,t) \,dt$ .
Proof. By Lemma 3.16, the limit on the left-hand side of (3.28) exists. By Lemma 3.3, the limit is the same as that of
where $t_0>0$ is arbitrarily large. In particular, this limit is non-negative. By Riesz representation theorem, it may be written as $\check {\mu }_{z}^{g, -}(\varphi )$ , where $\check {\mu }_{z}^{g, -}$ is a Radon measure on ${\mathbb X}_z^- \times {\mathbb R}$ . By Lemma 3.8, the marginal measure of $\check {\mu }_{z}^{g, -}$ on ${\mathbb R}$ under the natural projection map is the absolutely continuous measure $\check V^g(z,t) \,dt$ .
We define the Radon measure $\check {\mu }^g$ on ${\mathbb X} \times {\mathbb R}$ by setting, for any continuous compactly supported function $\varphi $ on ${\mathbb X} \times {\mathbb R}$ ,
The main result of this section is stated as follows.
Theorem 3.18. Let g be a Hölder continuous function on ${\mathbb X}$ such that $\nu (g) = 0$ and g is not a coboundary. Then, for any continuous compactly supported function $\varphi $ on ${\mathbb X} \times {\mathbb R}$ , we have
Moreover, the following harmonicity property holds:
Proof. We can assume that $\varphi $ is non-negative. By Lemma 3.17, for every $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ , we have
Thanks to the dominated convergence theorem, this will imply (3.29). Indeed, for $t \in {\mathbb R}$ , set $\theta (t) = \sup _{x \in {\mathbb X}} \varphi (x, t)$ , so that $\theta $ is a continuous compactly supported function on ${\mathbb R}$ . Note that
By Lemma 3.8, we have, uniformly in $z \in {\mathbb X}^+$ ,
By the dominated convergence theorem, we get (3.29).
Now we prove (3.30). By (3.29),
As $\nu $ is T-invariant, we have
By Lemma 3.3, the latter has the same limit, as $n \to \infty $ , as
We prove in the following that we can apply (3.29) to the function
to get
which proves (3.30).
To finish the proof, we need to show that (3.29) implies that for any continuous compact supported function $\varphi $ on ${\mathbb X} \times {\mathbb R}$ , as $n \to \infty $ , the quantity
converges to
This is a standard argument by an approximation. Indeed, for $\varepsilon>0$ and $t \in {\mathbb R}$ , set ${\chi _{\varepsilon }^-(t) = 0}$ if $t<0$ ; $\chi _{\varepsilon }^-(t) = {t}/{\varepsilon }$ if $0\leqslant t \leqslant \varepsilon $ and $\chi _{\varepsilon }^-(t) = 1$ if $t> \varepsilon $ . Define also $\chi _{\varepsilon }^+(t) = \chi _{\varepsilon }^-(t + \varepsilon )$ . Then, for any $n \geqslant 0$ , we have
By (3.30) and Lemma 3.3, we obtain
We claim that the left- and right-hand sides of the latter inequality converge to the integral in (3.31). Indeed, for $(x, t) \in {\mathbb X} \times {\mathbb R}$ , we have that $|\chi _{\varepsilon }^+(t) \varphi (x, t)|$ and $|\chi _{\varepsilon }^-(t) \varphi (x, t)|$ are dominated by $|\varphi (x, t)|$ . The conclusion now follows from the dominated convergence theorem.
As for the function $\check V^g$ , the measure $\check {\mu }^g$ enjoys the following continuity property on cohomology classes.
Lemma 3.19. Let $g \in {\mathscr B}$ with $\nu (g) = 0$ . Assume that g is not a coboundary. Let $\alpha \in (0,1)$ and $(h_m)_{m \geqslant 0}$ be a sequence of element of ${\mathscr B}_{\alpha }$ that converges to $0$ with respect to the Hölder norm $\|\cdot \|_{\alpha }$ . For $m \geqslant 0$ , set $g_m = g + h_m \circ T - h_m$ . Then, for any continuous compactly supported function $\varphi $ on ${\mathbb X} \times {\mathbb R}$ , we have
Proof. We can assume that $\varphi $ is non-negative. By Theorem 3.18, for $m \geqslant 0$ , we have
For any $n \geqslant 0$ , we have $S_n g_m \leqslant S_n g + 2 \|h_m\|_{\infty }$ . Hence, for $t \in {\mathbb R}$ , we have $\check {\tau }_t^{g_m} \leqslant \check {\tau }_{t + 2 \|h_m\|_{\infty }}^{g}$ . We obtain
Again by Theorem 3.18, as $n \to \infty $ , the latter quantity converges to
Thus, we have
In the same way, one also has
As $\varphi $ is continuous, the conclusion follows from the dominated convergence theorem.
Proof of Theorem 1.1.
So far we have proved Theorem 3.18 which is an analogue of Theorem 1.1 for the reversed dynamical system $({\mathbb X}, T^{-1}, \nu )$ . By Lemma 2.5, this dynamical system is isomorphic to a subshift of finite type equipped with a Gibbs measure. Therefore, Theorem 1.1 is actually equivalent to Theorem 3.18. Formally, the former can be obtained from the latter by replacing f with $g = f \circ T^{-1} \circ \iota = f \circ \iota \circ T$ , and $\nu $ with $\iota _* \nu $ .
The reader may note that (3.20) is a particular case of (1.3), which is the reason to call the Radon measure $\mu ^f$ harmonic.
4 Conditioned limit theorems
In this section we prove Theorems 1.3 and 1.5.
4.1 Proof of Theorem 1.3
As in the construction of the harmonic function $\check {V}^g$ and the harmonic measure $\check {\mu }^g$ , we prove Theorem 1.3 in several steps. The first step is to deal with the case of functions g depending only on the future. The following result follows from the general result for Markov chains established in [Reference Grama, Lauvergnat and Le Page15, Theorem 2.3]. The assumptions of this statement can be checked to hold thanks to the spectral gap properties of the Ruelle operator formulated in §5.1.
Lemma 4.1. Let $g \in {\mathscr B}^+$ with $\nu (g) = 0$ and assume that g is not a coboundary. Then, uniformly in $z \in {\mathbb X}^+$ and t in compact subsets of ${\mathbb R}$ ,
We have to strengthen Lemma 4.1 by proving the following integral form.
Lemma 4.2. Let $g \in {\mathscr B}^+$ with $\nu (g) = 0$ and assume that g is not a coboundary. Then, for any continuous compactly supported function $\varphi $ on ${\mathbb X}^-_z$ , we have, uniformly in $z \in {\mathbb X}^+$ and t in compact subsets of ${\mathbb R}$ ,
Proof. It suffices to prove this result when $\varphi $ is the indicator function of a cylinder set in ${\mathbb X}^-_z$ , because the general case follows by a standard approximation argument. Thus, let $m \geqslant 0$ and $a \in A_z^m$ and, as before, denote by $\mathbb C_{a,z}$ the associated cylinder in ${\mathbb X}^-_z$ (see (2.6)).
If $t + S_k g(T^{m-k} (a \cdot z)) \geqslant 0$ for every $1 \leqslant k \leqslant m$ , we have
By Lemma 4.1, as $n \to \infty $ , the latter quantity converges, uniformly in $z \in {\mathbb X}^+$ and t in compact subsets of ${\mathbb R}$ , to
which, by definition, is equal to $2 \check V^{g}(z, t) \check {\mu }^{g,-}_{z,t} (\mathbb C_{a,z}).$
If there exists $1 \leqslant k \leqslant m$ with $t + S_k g(T^{m-k} (a \cdot z)) <0$ , we have $\check {\mu }^{g,-}_{z,t} (\mathbb C_{a,z}) = 0$ and
for $n>k$ . The conclusion follows.
From Lemmas 4.1 and 4.2, we deduce the analogous result for functions which depend only on finitely many negative coordinates.
Lemma 4.3. Let $g \in {\mathscr B}$ be such that $\nu (g) = 0$ and there exists $m \geqslant 0$ with $g \circ T^m \in {\mathscr B}^+$ . Assume that g is not a coboundary. Then, uniformly in $z \in {\mathbb X}^+$ and t in compact subsets of ${\mathbb R}$ ,
Moreover, for any continuous compactly supported function $\varphi $ on ${\mathbb X}^-_z$ , uniformly in $z \in {\mathbb X}^+$ and t in compact subsets of ${\mathbb R}$ ,
Proof. As in Lemma 3.15, for $a \in A_z^m$ , let $\varphi _a$ be the continuous function $y \mapsto \varphi (a \cdot y)$ on ${\mathbb X}^-_{a \cdot z}$ . We have, by setting $h = g \circ T^m$ ,
The conclusion now follows from Lemmas 4.1 and 4.2 and (3.24).
Now we use the same approximation argument as before to deduce from Lemma 4.3 a slightly weaker statement that works for every function g in ${\mathscr B}$ . This is the main result of this section.
Theorem 4.4. Let $g \in {\mathscr B}$ be such that $\nu (g) = 0$ . Assume that g is not a coboundary. Then, for any continuous compactly supported function $\varphi $ on ${\mathbb R}$ , we have, uniformly in $z \in {\mathbb X}^+$ ,
Moreover, for any continuous compactly supported function $\varphi $ on ${\mathbb X} \times {\mathbb R}$ , we have
Proof. For $(z,t) \in {\mathbb X}^+ \times {\mathbb R}$ , denote
Let $(g_m)_{m \geqslant 0}$ be the sequence of Hölder continuous functions as in Lemma 2.11. For ${z \in {\mathbb X}^+}$ and $t \in {\mathbb R}$ , we have
By taking the limit as $n \to \infty $ , we get by Lemma 4.3,
The first part of the lemma now follows from Lemma 3.10.
Let now $\varphi $ be a non-negative continuous compactly supported function on ${\mathbb X} \times {\mathbb R}$ . For $m, n \geqslant 0$ , we have
Now we prove Theorem 1.3.
Proof of Theorem 1.3.
The first assertion of Theorem 1.3 follows from the second assertion of Theorem 4.4 by replacing f with $g = f \circ T^{-1} \circ \iota = f \circ \iota \circ T$ , and $\nu $ with $\iota _* \nu $ . The second assertion is also obtained from Theorem 4.4 by using Lemma 2.12.
From Theorem 1.3, we get the following coarse domination which will be used in the proof of the conditioned local limit theorem (Theorem 1.7).
Corollary 4.5. Let g be in ${\mathscr B}^+$ with $\nu (g) = 0$ . Assume that g is not cohomologous to $0$ . Let G be a continuous compactly supported function on ${\mathbb X}^+ \times {\mathbb R}$ . Then there exists a constant $c>0$ such that for any $n \geqslant 1$ ,
Proof. By replacing G with the function $\sup _{z \in {\mathbb X}^+} |G(z, t)|$ , we can assume that G does not depend on the first coordinate. Let $c_0$ be as in Lemma 2.9. For $t \in {\mathbb R}$ , set $G_1(t) = \sup _{|t' - t| \leqslant c_0} |G(t')|$ . Then for any $t \in {\mathbb R}$ and $z, z' \in {\mathbb X}^+$ with $z_0 = z_0'$ , we have
for some constant $c>0$ coming from Lemma 2.3. By integrating over $z' \in {\mathbb X}^+$ , we get
where $c_1 = \inf _{a \in A} \nu ^+ \{z' \in {\mathbb X}^+: z^{\prime }_0 = a \}$ . Integrating over $t \in {\mathbb R},$ we get the result by Theorem 1.3.
4.2 Proof of Theorem 1.5
Again we start with the case of Markov chains. As in the previous section, using the argument of [Reference Grama, Lauvergnat and Le Page15, Theorem 2.5], we get the following result.
Lemma 4.6. Let $g \in {\mathscr B}^+$ with $\nu (g) = 0$ and assume that g is not a coboundary. Then, for any continuous compactly supported function F on ${\mathbb X}^+ \times {\mathbb R}$ , we have, uniformly in $z \in {\mathbb X}^+$ and t in compact subsets of ${\mathbb R}$ ,
We extend the previous lemma to allow functions F depending on the past coordinates in ${\mathbb X}$ .
Lemma 4.7. Let $g \in {\mathscr B}^+$ with $\nu (g) = 0$ and assume that g is not a coboundary. Then, for any continuous compactly supported function F on ${\mathbb X} \times {\mathbb R}$ , we have, uniformly in $z \in {\mathbb X}^+$ and t in compact subsets of ${\mathbb R}$ ,
Proof. For $a \in A^{\{-m,\ldots ,-1\}}$ , set $\mathbb C_{a} = \{ x \in {\mathbb X} : x_{-m} = a_{-m}, \ldots , x_{-1} = a_{-1} \}.$ By a standard approximation argument, it suffices to prove the result for the set of functions F of the form
, where $F_1$ is a continuous compactly supported function on ${\mathbb X}^+ \times {\mathbb R}$ , and $a \in A^{\{-m,\ldots ,-1\}}$ with $M(a_{i-1}, a_{i}) = 1$ for $-m+1 \leqslant i \leqslant -1$ . We want to determine the limit as $n \to \infty $ ,
Note that in this integral, all the terms only depend on the coordinates $y_{-n}, y_{-n+1}, \ldots , y_{-1}$ except $T^{-n}(y \cdot z)$ . By integrating first over the deep past coordinates $\ldots , y_{-n-2}, y_{-n-1}$ , we get by using Lemma 2.2,
where, for $(z',t) \in {\mathbb X}^+ \times {\mathbb R}$ ,
Lemma 4.6 gives uniformly in $z \in {\mathbb X}^+$ and t in compact subsets of ${\mathbb R}$ ,
By construction of the measure $\nu $ in Lemma 2.2, we have
which ends the proof of the lemma.
As for Theorem 1.3, we get the following version of Lemma 4.7, where we add a source target function.
Lemma 4.8. Let $g \in {\mathscr B}^+$ with $\nu (g) = 0$ and assume that g is not a coboundary. Then, for any continuous compactly supported function F on ${\mathbb X}_z^- \times {\mathbb X} \times {\mathbb R}$ , we have, uniformly in $z \in {\mathbb X}^+$ and t in compact subsets of ${\mathbb R}$ ,
The proof of Lemma 4.8 can be carried out in the same way as that of Lemma 4.2 and therefore is left to the reader. By using again conditioning and Lemma 3.15, we extend the previous lemma to functions g depending on finitely many coordinates of the past.
Lemma 4.9. Let $g \in {\mathscr B}$ be such that $\nu (g) = 0$ and there exists $m \geqslant 0$ with $g \circ T^m \in {\mathscr B}^+$ . Assume that g is not a coboundary. Then, for any continuous compactly supported function F on ${\mathbb X} \times {\mathbb R}$ , we have, uniformly in $z \in {\mathbb X}^+$ and t in compact subsets of ${\mathbb R}$ ,
Moreover, for any continuous compactly supported function F on ${\mathbb X}^-_z \times {\mathbb X} \times {\mathbb R}$ , we have, uniformly in $z \in {\mathbb X}^+$ and t in compact subsets of ${\mathbb R}$ ,
Proof. We prove only the second assertion, because the first is a particular case of the second. As in Lemma 3.15, for $a \in A_z^m$ , set $F_a$ to be the function on ${\mathbb X}^-_{a \cdot z} \times {\mathbb X} \times {\mathbb R}$ defined by $F_a (y, x, t) = F (y \cdot a, T^m x, t)$ . We have, by setting $h = g \circ T^m$ ,
The same technique as in Lemma 4.4 gives the following result.
Lemma 4.10. Let $g \in {\mathscr B}$ with $\nu (g) = 0$ and assume that g is not a coboundary. Then, for any continuous compactly supported function F on ${\mathbb X} \times {\mathbb X} \times {\mathbb R} \times {\mathbb R}$ , we have
5 Effective local limit theorems
So far we have adapted some results from the theory of Markov chains to the case of hyperbolic dynamical systems by constructing the analogues of the harmonic functions $V^g$ and $\check V^g$ and building the harmonic measures $\mu ^g$ and $\check {\mu }^g$ . In the remaining part of the paper, we use these objects to establish conditioned limit theorems, by adapting the strategy from the case of sums of independent random variables [Reference Grama and Xiao18]. We start with formulating an effective version of the ordinary local limit theorem which is adapted to our needs.
5.1 Spectral gap theory
Fix $\alpha \in (0,1)$ such that $\psi \in {\mathscr B}^+_{\alpha }$ , where $\psi $ is the potential function used for the construction of the Gibbs measure $\nu $ (see §2.1). Denote by $\mathscr L({\mathscr B}^+_{\alpha }, {\mathscr B}^+_{\alpha })$ the set of all bounded linear operators from ${\mathscr B}^+_{\alpha }$ to ${\mathscr B}^+_{\alpha }$ equipped with the standard operator norm $\| \cdot \|_{{\mathscr B}^+_{\alpha } \to {\mathscr B}^+_{\alpha }}$ . From the general construction of the Ruelle operator, every $ f \in {\mathscr B}^+_{\alpha }$ gives rise to a family of perturbed operators $(\mathcal {L}_{\psi + \mathbf {i} t f})$ defined as follows: for any $\varphi \in {\mathscr B}^+_{\alpha }$ ,
By iteration, it follows that for any $\psi , f \in {\mathscr B}_{\alpha }$ and $t \in {\mathbb R}$ ,
The following result (see [Reference Parry and Pollicott24]) provides the spectral gap properties for the perturbed operator $\mathcal {L}_{\psi + \mathbf {i} t f}$ . For similar statements in the case of Markov chains we refer to [Reference Hennion and Hervé21].
Lemma 5.1. Assume that $f\in {\mathscr B}^+_{\alpha }$ is not a coboundary and that $\nu (f)=0$ . Then, there exists a constant $\delta> 0$ such that for any $t \in (-\delta , \delta )$ ,
where the mappings $t \mapsto \Pi _{ t}: (-\delta , \delta ) \to \mathscr L({\mathscr B}^+_{\alpha }, {\mathscr B}^+_{\alpha })$ and $z \mapsto N_{ t}: (-\delta , \delta ) \to \mathscr L({\mathscr B}^+_{\alpha }, {\mathscr B}^+_{\alpha })$ are analytic in the operator norm topology, $\Pi _{ t}$ is a rank-one projection with $\Pi _{0}(\varphi )(z) = \nu ^+ (\varphi )$ for any $\varphi \in \mathscr {B}^+_{\alpha }$ and $z \in {\mathbb X}^+$ , $\Pi _{ t} N_{ t} = N_{ t} \Pi _{ t} = 0$ . Moreover, there exist $n_0\geqslant 1$ and $q \in (0,1)$ such that for any $t\in (-\delta ,\delta )$ the $\|N_{t}^{n_0}\|_{{\mathscr B}^+_{\alpha } \to {\mathscr B}^+_{\alpha }} \leqslant q$ .
The eigenvalue $\unicode{x3bb} _{t}$ has the asymptotic expansion: as $t \to 0$ ,
Note that because f is not a coboundary with respect to T, the asymptotic variance $\sigma _f^2$ appearing in (5.3) is strictly positive.
Lemma 5.2. Let $f\in {\mathscr B}^+_{\alpha }$ and $t \neq 0$ . Assume that for any $p \neq 0$ and $q \in {\mathbb R}$ , the function $p f + q$ is not cohomologous to a function with values in ${\mathbb Z}$ . Then, for any $t \neq 0$ , the operator $\mathcal L_{\psi +{\mathbf {i}} t f }$ has spectral radius strictly less than $1$ in ${\mathscr B}^+_{\alpha }$ . More precisely, for any compact set $K \subset {\mathbb R} \setminus \{0\}$ , there exist constants $c_K, c_K'>0$ such that for any $\varphi \in \mathscr {B}^+_{\alpha }$ and $n \geqslant 1$ ,
Proof. The proof of the first assertion can be found in [Reference Parry and Pollicott24, Theorem 4.5]. Now we prove (5.4). For every $t \in K$ , there exist $n_0(t) \geqslant 1$ and $\alpha (t) \in (0,1)$ such that $\|\mathcal L_{\psi +{\mathbf {i}}t f }^{n_0(t)} \|_{\mathscr {B}^+_{\alpha } \to \mathscr {B}^+_{\alpha }} < \alpha (t)$ . As the operator $\mathcal L_{\psi +{\mathbf {i}}t f }$ depends continuously on t for the operator norm topology, there exists $\delta = \delta (t)$ such that for any $s \in (t - \delta (t), t + \delta (t))$ , we still have $\|\mathcal L_{\psi +{\mathbf {i}}s f }^{n_0(t)} \|_{\mathscr {B}^+_{\alpha } \to \mathscr {B}^+_{\alpha }} < 1$ . In particular, for every $n \geqslant 0$ we have $\|\mathcal L_{\psi +{\mathbf {i}}s f }^{n} \|_{\mathscr {B}^+_{\alpha } \to \mathscr {B}^+_{\alpha }} \leqslant c(t) \alpha (t)^{n/n_0(t)}$ , for some $c(t)> 0$ . By compactness, we can find $t_1, \ldots , t_r \in K$ such that $K \subset \bigcup _{i =1}^r (t_i - \delta (t_i), t_i + \delta (t_i))$ . In particular, by setting $c = \max _{1 \leqslant i \leqslant r} c(t_i)$ , $\alpha = \max _{1 \leqslant i \leqslant r} \alpha (t_i)$ and $n_0 = \max _{1 \leqslant i \leqslant r} n_0(t_i)$ , we get for any $s \in K$ and $n \geqslant 0$ , $\|\mathcal L_{\psi +{\mathbf {i}}s f }^{n} \|_{\mathscr {B}^+_{\alpha } \to \mathscr {B}^+_{\alpha }} \leqslant c \alpha ^{n/n_0}$ .
5.2 Local limit theorem for smooth target functions
In the following we establish a local limit theorem for Markov chains with a precise estimation of the remainder term. Let F be a measurable non-negative bounded target function on ${\mathbb X} \times \mathbb {R}$ . The probability we are interested in can be written as follows: for any $z \in {\mathbb X}^+$ ,
The main difficulty is to give a local limit theorem with the explicit dependence of the remainder terms on F.
We first describe the kind of target functions that we will use.
Lemma 5.3. Let X be a compact metric space and $\alpha>0$ . Let F be a real-valued function on $X \times {\mathbb R}$ such that:
-
(1) for any $t \in {\mathbb R}$ , the function $z \mapsto F(z, t)$ is $\alpha $ -Hölder continuous on X;
-
(2) for any $z \in X$ , the function $t \mapsto F(z, t)$ is measurable on ${\mathbb R}$ .
Then, the function $(z, t) \mapsto F(z,t)$ is measurable on $X \times {\mathbb R}$ and the function ${t \mapsto \| F (\cdot , t) \|_{\alpha }}$ is measurable on ${\mathbb R}$ , where the norm $\|\cdot \|_{\alpha }$ is the usual norm on the space of $\alpha $ -Hölder continuous functions on X. Moreover, if the integral $\int _{{\mathbb R}} \| F (\cdot , t) \|_{\alpha } \,dt$ is finite, we define the partial Fourier transform $\widehat F$ of F by setting for any $z \in X$ and $u \in {\mathbb R}$ ,
This is a continuous function on $X \times {\mathbb R}$ . In addition, for every $u \in {\mathbb R}$ , the function ${z \mapsto \widehat F(z, u)}$ is $\alpha $ -Hölder continuous and $\| \widehat F (\cdot , u) \|_{\alpha } \leqslant \int _{{\mathbb R}} \| F (\cdot , t) \|_{\alpha } \,dt$ .
Proof. As the space X is separable and the function $z \mapsto F (z, t)$ is continuous on X for any $t \in {\mathbb R}$ , the supremum $\sup _{z\in X} | F(z,t) |$ can be taken over a countable dense subset, so that $t \mapsto \sup _{z\in X} | F(z,t) |$ is measurable. In the same way, because the function $z \mapsto F (z, t)$ is $\alpha $ -Hölder continuous on X for any $t \in {\mathbb R}$ , one can also verify that $\sup _{z, z' \in X} ({| F (z, t ) - F (z', t ) |}/{\alpha ^{\omega (z, z')}})$ is a measurable function in t.
In case the integral $\int _{{\mathbb R}} \| F (\cdot , t) \|_{\alpha } \,dt$ is finite, the partial Fourier transform $\widehat F$ is well defined and continuous by the dominated convergence theorem. The norm domination is obvious.
We denote by $\mathscr H_{\alpha }^{\ \, +}$ the set of real-valued functions on ${\mathbb X}^+ \times {\mathbb R}$ such that conditions (1) and (2) of Lemma 5.3 hold and the integral $\int _{{\mathbb R}} \| F (\cdot , t) \|_{{\mathscr B}^+_{\alpha }} \,dt$ is finite. For any compact set $K\subset {\mathbb R} $ , denote by $\mathscr H_{\alpha , K}^{\ \, +}$ the set of functions $F \in \mathscr H_{\alpha }^{\ \, +}$ such that the Fourier transform $\widehat F(z,\cdot )$ has a support contained in K for any $z \in {\mathbb X}^+$ . Let $\phi $ be the standard normal density:
Theorem 5.4. Let $\alpha \in (0,1)$ . Assume that $g \in {\mathscr B}^+_{\alpha }$ such that $\nu ^+ (g)=0$ and for any $p \neq 0$ and $q \in {\mathbb R}$ , the function $p g + q$ is not cohomologous to a function with values in ${\mathbb Z}$ . Let $K \subset {\mathbb R}$ be a compact set. Then there exists a constant $c_K>0$ such that for any $F\in \mathscr H_{\alpha , K}^{\ \, +}$ , $n\geqslant 1$ and $z \in {\mathbb X}^+$ ,
Proof. Without loss of generality, we assume that $\sigma _g = 1$ . By the Fourier inversion formula, the Fubini theorem and a change of variable t to $({t}/{\sqrt {n}})$ , we get
Note that the Fubini theorem can be applied because the integral on ${\mathbb X}^-_z$ is, in fact, a finite sum. Denote
We decompose the integral I into three parts: $I=I_{1}+I_{2} + I_3,$ where
Estimate of $I_1.$ As $\int _{{\mathbb R}} \|F(\cdot , u)\|_{{\mathscr B}^+_{\alpha }} \,du < \infty $ , the function $z \mapsto \widehat {F} (z, t )$ is Hölder continuous on ${\mathbb X}^+$ with Hölder norm at most $\int _{{\mathbb R}} \|F(\cdot , u)\|_{{\mathscr B}^+_{\alpha }} \,du$ , for any fixed $t \in {\mathbb R}$ . Applying (5.2), we get
For the first term, by (5.3) and simple calculations, we get
For the second and third terms, using again Lemma 5.1, we obtain
Therefore, we obtain the following upper bound for $I_1$ :
Estimate of $I_2.$ As the function $\widehat {F}(z,\cdot )$ is compactly supported on $K \subset [-C_1, C_1 ]$ , where $C_1>0$ is a constant not depending on $z \in {\mathbb X}^+$ , we have
Note that, for any t satisfying $\delta < |t| \leqslant C_1 $ ,
Then, by Lemma 5.2, it follows that
Estimate of $I_3.$ Note that
For the first term, by the Fourier inversion formula,
For the second term, using the fact that $\widehat {F} (z, {t}/{\sqrt {n}} ) \leqslant \int _{{\mathbb R}} | F (z, u) | \,du$ , we have
Combining (5.6), (5.7), (5.8) and (5.9), and taking into account that $\delta $ is a fixed constant, we conclude the proof of (5.5).
5.3 Local limit theorem for $\varepsilon $ -dominated target functions
Let $\varepsilon>0$ . Let $f, g$ be functions on ${\mathbb R}$ . We say that the function $g \ \varepsilon $ -dominates the function f (or $f \ \varepsilon $ -minorates g) if for any $t \in {\mathbb R}$ , it holds that
In this case, we write $f \leqslant _{\varepsilon } g$ or $g \geqslant _{\varepsilon } f$ . For any functions F and G on ${\mathbb X}^+ \times {\mathbb R}$ , we say that $F \leqslant _{\varepsilon } G$ if $F(z, \cdot ) \leqslant _{\varepsilon } G(z, \cdot )$ for any $z \in {\mathbb X}^+$ .
In the proofs we make use of the following assertion. Denote by $\rho $ the non-negative density function on ${\mathbb R}$ , which is the Fourier transform of the function for $t \in {\mathbb R}$ . Set $\rho _{\varepsilon }(u) = ({1}/{\varepsilon }) \rho ( {u}/{\varepsilon } )$ for $u \in {\mathbb R}$ and $\varepsilon>0$ .
Lemma 5.5. Let $\varepsilon \in (0,1/4)$ . Let $f: {\mathbb R} \to {\mathbb R}_+$ and $g: {\mathbb R} \to {\mathbb R}_+$ be integrable functions satisfying $f\leqslant _{\varepsilon } g$ . Then, for any $u \in {\mathbb R}$ ,
Remark 5.6. The domination property $\leqslant _{\varepsilon } $ implies, in particular, that if $f \leqslant _{\varepsilon } g$ and the function g is integrable, then f is bounded and $\lim _{u\to \infty } f(u)=0$ , ${\lim _{u\to -\infty } f(u)=0}$ . Indeed, because $f \leqslant _{\varepsilon } g$ and g is an integrable function, by Lemma 5.5 we have $f \leqslant ({1}/({1 - 2 \varepsilon })) g * \rho _{\varepsilon ^2}$ . As the Fourier transform of $g * \rho _{\varepsilon ^2}$ is compactly supported on $[- {1}/{\varepsilon ^2}, {1}/{\varepsilon ^2}]$ , by the Fourier inversion formula,
Therefore, $g * \rho _{\varepsilon ^2}$ is bounded on ${\mathbb R}$ , so that f is bounded on ${\mathbb R}$ .
In the following, for any function $F \in \mathscr H_{\alpha }^{\ \,+}$ , we use the notation
and
The following properties are useful in the proofs.
Lemma 5.7. Let $F\in \mathscr H^{\ \,+}_{\alpha } $ and $\rho \in L^1({\mathbb R})$ . Then $F*\rho \in \mathscr H^{\ \,+}_{\alpha } $ and $\| F *\rho \|_{\mathscr H^{\ \,+}_{\alpha }} \leqslant \| F \|_{\mathscr H^{\ \,+}_{\alpha }} \| \rho \|_{L^1({\mathbb R})}$ .
Theorem 5.8. Let $\alpha \in (0, 1)$ and $g\in {\mathscr B}^+_{\alpha }$ be such that $\nu ^+ (g)=0$ . Assume that for any $p \neq 0$ and $q \in {\mathbb R}$ , the function $p g + q$ is not cohomologous to a function with values in ${\mathbb Z}$ . There exists $c>0$ with the following property: for any $\varepsilon \in (0, \tfrac 18)$ , there exists a constant $c_{\varepsilon }>0$ such that for any non-negative function F and any function $G \in \mathscr H_{\alpha }^{\ \,+}$ satisfying $F \leqslant _{\varepsilon } G$ , $n\geqslant 1$ and $z \in {\mathbb X}^+$ ,
and for any non-negative function F and non-negative functions $G, H \in \mathscr H_{\alpha }^{\ \,+}$ satisfying $H \leqslant _{\varepsilon } F \leqslant _{\varepsilon } G$ , $n\geqslant 1$ and $z \in {\mathbb X}^+$ ,
Proof. Without loss of generality, we assume that $\sigma _g =1$ . We first prove the upper bound (5.10). By Lemma 5.5, we have $F \leqslant (1 + 4 \varepsilon ) G * \rho _{\varepsilon ^2}$ and, hence,
By Lemma 5.7, $\widehat {G * \rho _{\varepsilon ^2}}\in \mathscr H^{\ \,+}_{\alpha }$ , and the support of the function $\widehat {G * \rho _{\varepsilon ^2}} (z, \cdot ) = \widehat {G} (z, \cdot ) \widehat {\rho }_{\varepsilon ^2}(\cdot )$ . is included in $[- {1}/{\varepsilon ^2}, {1}/{\varepsilon ^2}]$ , for all $z \in {\mathbb X}^+$ . Using Theorem 5.4, for any $\varepsilon \in (0, \tfrac 14)$ , there exists $c_{\varepsilon }>0$ such that for all $n\geqslant 1$ and $z \in {\mathbb X}^+$ ,
By a change of variable and Fubini’s theorem, we have for any $z \in {\mathbb X}^+$ ,
where $\phi _{\sqrt {n}}(t)\hspace{-0.5pt} =\hspace{-0.5pt} ({1}/{\sqrt {2 \pi n}}) e^{- {t^2}/{2n}}$ , $t\hspace{-0.5pt} \in\hspace{-0.5pt} {\mathbb R}$ . For brevity, denote $\psi (t)\hspace{-0.5pt}=\hspace{-0.5pt}\sup _{|v|\leqslant \varepsilon } \phi _{\sqrt {n}} (t\hspace{-0.5pt} +\hspace{-0.5pt} v)$ , $t \in {\mathbb R}$ . Using the second inequality in Lemma 5.5, we have
For $J_1$ , by Taylor’s expansion and the fact that the function $\phi '$ is bounded on ${\mathbb R}$ , we derive that
For $J_2$ , because $\phi _{\sqrt {n}} \leqslant {1}/{\sqrt {n}}$ and $\int _{|v| \geqslant \varepsilon } \rho _{\varepsilon ^2} (v) \, d v \leqslant 2 \varepsilon $ , we get
From (5.15) and (5.16), together with (5.12) and (5.13), we get (5.10).
Now we prove the lower bound (5.11). As $F \geqslant _{\varepsilon } H$ , using the second inequality in Lemma 5.5, we get
For the first term, by Theorem 5.4, for any $\varepsilon>0$ , there exists $c>0$ such that for all $n\geqslant 1$ and $z \in {\mathbb X}^+$ ,
In the same way as in (5.14), we have
Using the first inequality in Lemma 5.5, we have $\phi _{\sqrt {n}} *\rho _{\varepsilon ^2} (t) \geqslant (1 - 2 \varepsilon ) \psi (t)$ , for $t\in {\mathbb R}$ , where $\psi (t) = \inf _{|v| \leqslant \varepsilon } \phi _{\sqrt {n}} (t + v)$ . Proceeding in the same way as in (5.15) and (5.16), we obtain that
For the second term on the right-hand side of (5.17), using (5.10) and the fact that $H \leqslant _{2\varepsilon } G$ and $\phi \leqslant 1$ , we get that there exist constants $c, c_{\varepsilon }>0$ such that for any $v \in {\mathbb R}$ and $n \geqslant 1$ ,
This, together with the fact that $\int _{|v| \geqslant \varepsilon } \rho _{\varepsilon ^2} (v) d v \leqslant 2 \varepsilon $ , implies
From (5.17), (5.18), (5.20) and (5.21), we obtain the lower bound (5.11).
6 Effective conditioned local limit theorems
6.1 Formulation of the result
We prove the following conditioned local limit theorem for Markov chains which provides a rate of order $n^{-1}$ . This result will serve as an intermediate step between the conditioned central limit Theorem 1.5 and the conditioned local limit Theorem 1.7. The interest of this result lies in the fact that it is uniform in the function F. In particular, the theorem is effective when the support of the function F moves to infinity with the rate $\sqrt {n}$ . This strategy is inspired by [Reference Denisov and Wachtel7] for random walks in cones of ${\mathbb R}^d$ , see also [Reference Grama, Lauvergnat and Le Page16] for finite Markov chains and [Reference Grama and Xiao18] for random walks on ${\mathbb R}$ . For a different approach based on the Wiener–Hopf factorisation we refer to [Reference Caravenna6, Reference Doney10, Reference Vatutin and Wachtel32].
Theorem 6.1. Let $\alpha \in (0,1)$ and $g \in {\mathscr B}^+_{\alpha }$ be such that $\nu ^+ (g) = 0$ . Assume that for any $p \neq 0$ and $q \in {\mathbb R}$ , the function $p g + q$ is not cohomologous to a function with values in ${\mathbb Z}$ . Let $t_0 \in {\mathbb R}_+$ . Then, there exist a constant $c>0$ and a sequence $(r_n)$ of positive numbers satisfying $\lim _{n \to \infty } r_n = 0$ with the following properties.
-
(1) For any $\varepsilon \in (0,\tfrac 18)$ , there exists a constant $c_{\varepsilon }> 0$ such that for any $n \geqslant 1$ , $z \in {\mathbb X}^+$ , $t \leqslant t_0$ , any functions $F, G: {\mathbb X}^+ \times {\mathbb R} \to {\mathbb R}_+$ satisfying $F \leqslant _{\varepsilon } G$ , $G \in \mathscr H_{\alpha }^{\ \,+}$ ,
(6.1) -
(2) For any $\varepsilon \in (0,\tfrac 18)$ , there exists a constant $c_{\varepsilon }> 0$ such that for any $n \geqslant 1$ , $z \in {\mathbb X}^+$ , $t \leqslant t_0$ , any functions $F, G, H: {\mathbb X}^+ \times {\mathbb R} \to {\mathbb R}_+$ satisfying $H \leqslant _{\varepsilon } F \leqslant _{\varepsilon } G$ , ${G, H \in \mathscr H_{\alpha }^{\ \,+}}$ ,
(6.2)
6.2 Preparatory statements
The normal density of variance $v> 0$ is denoted by
and the Rayleigh density with scale parameter $\sqrt {v}$ is denoted by
The standard normal density is denoted by $\phi (x) = \phi _1(x)$ , $x\in {\mathbb R}$ . The following lemma from [Reference Grama and Xiao18] shows that when v is small the convolution $\phi _{v} * \phi ^+_{1-v}$ behaves like the Rayleigh density.
Lemma 6.2. For any $v \in (0,1/2]$ and $x\in {\mathbb R}$ , it holds
We need the following inequality of Haeusler [Reference Haeusler19, Lemma 1], which is a generalisation of Fuk’s inequality for martingales.
Lemma 6.3. Let $\xi _1, \ldots , \xi _n$ be a martingale difference sequence with respect to the non-decreasing $\sigma $ -fields $\mathscr F_0, \mathscr F_1, \ldots , \mathscr F_n$ . Then, for all $u, v, w> 0$ ,
Using this lemma we establish the following Fuk-type inequality involving a target function on the Markov chain $((T^{-n} y\cdot z)_+)_{n\geqslant 0}$ .
Lemma 6.4. Let $\alpha \in (0,1)$ , $g\in {\mathscr B}^+_{\alpha }$ such that $\nu ^+(g) = 0$ and $\varphi \in {\mathscr B}^+_{\alpha }$ be non-negative. There exist constants $c, c', c_0>0$ such that for any $M> c_0$ and $n\geqslant 1 $ ,
Proof. By Propositions 2.6 and 2.7 and Lemma 2.8, there exists a Hölder continuous function $g_0$ on ${\mathbb X}^+$ satisfying $\mathcal L_{\psi } g_0 = 0$ such that $\{ y\mapsto \check S_k g_0 ( (y \cdot z)_+) \}_{k \geqslant 0}$ is a martingale on ${\mathbb X}_z^-$ and $\sup _{k \geqslant 0} \| \check S_k g_0 - \check S_k g \|_{\infty } \leqslant c$ for some constant $c>0$ . Let ${c_0=1+\|g_0\|_{\infty } < \infty }$ . In addition, with $p=n-[n^{1/3}]$ , we have $\max _{1 \leqslant j \leqslant n } | \check S_{j} g_0(x') | \leqslant \max _{1 \leqslant j \leqslant p } | \check S_{j} g_0(x') | + c_0 n^{1/3}$ . With these properties, it holds that, for n large enough,
Note that $\mathcal L^k_{\psi } \varphi (z) = \int _{{\mathbb X}^-_z} \varphi ( (T^{-k} y \cdot z)_+) \nu ^-_z(dy)$ with $k\geqslant 1$ , where $\mathcal L^k_{\psi }$ is defined by (2.3). Moreover, by Lemma 5.1 with $t=0$ , for $k\geqslant 1$ ,
By the Markov property, we have that for any $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ ,
We apply Fuk’s inequality for martingales (Lemma 6.3) with $u = \tfrac 12M\sqrt { n} $ , $v = c_0^2 \sqrt {n}$ and $w= ({c_0^2}/{8}) M n$ , so that the second and the third terms in the right-hand side of (6.3) vanish. This gives
This ends the proof of the lemma.
In order to control certain natural quantities appearing in the proof, we need the following definitions. For $\varepsilon>0 $ ,
Denote $\overline \chi _{\varepsilon }(u) = 1 - \chi _{\varepsilon }(u)$ and note that
Lemma 6.5. Let $\alpha \in (0,1)$ and $g \in {\mathscr B}^+_{\alpha }$ be such that $\nu ^+ (g) = 0$ . Assume that g is not a coboundary. Let $\kappa $ be a smooth compactly supported function on ${\mathbb R}$ and $\varepsilon>0$ . Then there exists a constant $c>0$ such that for any $G \in \mathscr H^{\ \,+}_{\alpha }$ and any $m \geqslant 1$ , the function $A_m$ defined on ${\mathbb X}^+ \times {\mathbb R}$ by
belongs to $\mathscr H^{\ \,+}_{\alpha }$ and satisfies
Proof. For the first inequality, we write
which gives
This finishes the proof of the first inequality.
For the second inequality, recall that
We pick $c_0>0$ as in Lemma 2.9 and for $t \in {\mathbb R}$ we set $\kappa _1(t) = \sup _{|s| \leqslant c_0} |\kappa (t+s)|$ and $H(t) = \sup _{z \in {\mathbb X}^+} |G(z, t)|$ . We get for $z, z' \in {\mathbb X}^+$ with $z_0 = z_0'$ and $t \in {\mathbb R}$ ,
By Lemma 2.3, we get
for some constant c. By integrating over $z'$ , we get
By integrating over t, it follows that
Now we dominate the second term in the norm $\| A_{m} \|_{\mathscr H^{\ \,+}_{\alpha }}$ . For $t \in {\mathbb R}$ , set $\kappa _2(t) = \sup _{|s| \leqslant c_0} |\kappa '(t+s)|$ , where $c_0$ is the constant from Lemma 2.9. We get for $|t-t'| \leqslant c_0$ and $z \in {\mathbb X}^+$ ,
Hence, for $z, z', z" \in {\mathbb X}^+$ with $z_0 = z_0' = z_0"$ and $t \in {\mathbb R}$ ,
where $\kappa _3(t) = \sup _{|s| \leqslant c_0} |\kappa _2(t+s)|$ with $c_0$ from Lemma 2.9; for the second inequality we have applied Lemma 2.9 and for the last inequality we have used Lemma 2.3. Again by integrating over $z"$ , we get that
In addition, as G is in $\mathscr H_{\alpha }^{\ \,+}$ , the function $L(t) = \sup _{z, z' \in {\mathbb X}^+} \alpha ^{-\omega (z.z')} |G(z,t) - G(z',t)|$ is integrable on ${\mathbb R}$ and for $z,z' \in {\mathbb X}^+$ with $z_0 = z_0'$ and $t \in {\mathbb R}$ , we have
where we have again used Lemmas 2.3 and 2.9.
As $\overline \chi _{\varepsilon }$ is $1/\varepsilon $ -Lipschitz continuous on ${\mathbb R}$ , by reasoning in the same way and using Corollary 2.10, we get
By Lemma 2.3, we have
Putting (6.8), (6.9), (6.10) and (6.11) together, and integrating over $t\in {\mathbb R}$ , yields the required domination.
6.3 Proof of the upper bound
We prove the inequality (6.1) in Theorem 6.1. It is enough to prove (6.1) only for sufficiently large $n>n_0(\varepsilon )$ , where $n_0(\varepsilon )$ depends on $\varepsilon $ , otherwise the bound becomes trivial.
Without loss of generality, we assume that $\sigma _g = 1$ . Let $\varepsilon \in (0,\tfrac 18)$ . With $\delta = \sqrt {\varepsilon }$ , set $m=[ \delta n ]$ and $k = n-m$ . Note that $\tfrac 12\delta \leqslant {m}/{k} \leqslant {\delta }/({1-\delta })$ for $n \geqslant {2}/{\sqrt {\varepsilon }}$ . Denote, for $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ ,
By the Markov property, we have that for any $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ ,
By bounding the indicator function by
in the definition of $\Psi _m$ , we get
Let $G_{\varepsilon } (z, u)=G (z, u) \chi _{\varepsilon } (u-\varepsilon )$ for $z \in {\mathbb X}^+$ and $u\in {\mathbb R}$ , where $\varepsilon \in (0,1)$ and $\chi _{\varepsilon }$ is defined in (6.6). By the local limit theorem (cf. Theorem 5.8), there exist constants $c, c_{\varepsilon }>0$ such that for any $m \geqslant 1$ , $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ ,
where, for brevity, we set
Using (6.12), (6.13) and (6.14), and Lemma 4.1, we get that uniformly in $z \in {\mathbb X}^+$ and $t \leqslant t_0$ ,
Now we deal with the first term on the right-hand side of (6.16). Denote ${L_m(s) = H_m(\sqrt {k} s)}$ for $s \in {\mathbb R}$ . We have
As the function $s \mapsto L_m(s)$ is differentiable on ${\mathbb R}$ and vanishes as $s \to - \infty $ , using integration by parts, we have, for any $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ ,
Applying the conditioned central limit theorem (see Lemma 4.6), we have
where $r_k \to 0$ as $k \to \infty $ and by $\Phi ^+$ we denoted the Rayleigh cumulative distribution function (1.7). By (6.17), we have
By integration by parts and a change of variable, we have
Hence, from (6.18), (6.19) and (6.20), we obtain
Implementing this bound into (6.16) and using the fact that $\| G_{\varepsilon } \|_{\nu ^+\otimes \operatorname {\mathrm {Leb}}} \leqslant \| G \|_{\nu ^+\otimes \operatorname {\mathrm {Leb}}}$ and $ \| G_{\varepsilon } \|_{\mathscr H^{\ \,+}_{\alpha }} \leqslant \| G \|_{\mathscr H^{\ \,+}_{\alpha }}$ , we get, uniformly in $z \in {\mathbb X}^+$ and $t\leqslant t_0$ ,
where
By the definition of $H_m$ (cf. (6.15)) and Fubini’s theorem, it follows that
Denote $\delta _n = {m}/{n} = {[\delta n]}/{n}$ . By a change of variable, we have
where in the last line we used the fact that $G_{\varepsilon }(z, u)=0$ for any $z \in {\mathbb X}^+$ and $u\leqslant -\varepsilon $ . We handle the convolution $\phi _{\delta _n}*\phi _{1-\delta _n}^+$ using Lemma 6.2 together with the fact that $\delta _n={m}/{n}$ , $1 - \delta _n = {k}/{n}$ and $u \geqslant -\varepsilon $ :
As $G_{\varepsilon } \leqslant G$ , it follows that
Substituting this into (6.22), and using the fact that $\check V^{g}(z, t) \leqslant t + c$ gives
As $\varepsilon ^{1/2} n \geqslant m\geqslant \tfrac 12\varepsilon ^{1/2} n$ and $n> k\geqslant \tfrac 12n$ , we obtain
which finishes the proof of the upper bound (6.1).
6.4 Proof of the lower bound
We now proceed to prove the second assertion (6.2) of Theorem 6.1. We use the same notation as that in the proof of the upper bound. Recall that $\delta =\sqrt {\varepsilon }$ , $m=[\delta n]$ and $k=n-m$ . For $z \in {\mathbb X}^+$ , $t \in {\mathbb R}$ and $n \geqslant 1$ , denote
Note that $\Psi _n (z,t) = 0$ for $t \leqslant - c_0= -\|g\|_{\infty }$ because
for these values of t, and therefore in the following we can consider that $t\leqslant t_0$ . By the Markov property, we have that for any $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ ,
We write $\Psi _m$ as a sum of two terms: for any $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ ,
where
This implies that for any $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ ,
where
We proceed to give a lower bound for the term $J_n(z,t)$ in (6.28). It can be handled as the case of the upper bound, but here the situation is more complicated. By the local limit theorem (cf. Theorem 5.8), we get that there exist constants $c, c_{\varepsilon }>0$ such that for any $m \geqslant 1$ , $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ ,
where, for brevity, we set
Using (6.26), (6.29) and (6.31), and Lemma 4.1, we get that for any $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ ,
For the first term on the right-hand side of (6.33), proceeding in the same way as that in the proof of (6.21), using the lower bound in the conditioned central limit theorem (see Lemma 4.6), one can verify that
Implementing this bound into (6.33), we get that for any $z \in {\mathbb X}^+$ ,
where
In the same way as in the proof of (6.23), we have
where in the last inequality we applied Lemma 6.2 and the fact that $\phi ^+(u) = 0$ for $u < 0$ . Substituting this into (6.35), and using the fact that $H \leqslant _{\varepsilon } G$ and $\check V^{g}(z, t) \leqslant t + c$ , we get
As $\sqrt {{n}/{k}}\leqslant 1+ c \varepsilon ^{1/4}$ , $m\geqslant \tfrac 12 \varepsilon ^{1/2} n$ and $k\geqslant \tfrac 12n$ , using again $H \leqslant _{\varepsilon } G$ we deduce that for n sufficiently large,
We now deal with $K_n(z,t)$ which is the second term in (6.28). Bounding $K_n(z,t)$ is one of the difficult points of the paper and needs to make use of the duality. We start by splitting $K_n(z,t)$ into two parts according to whether the values of $t + \check S_k g(y \cdot z)$ are less or larger than $\varepsilon \sqrt {n}$ : for $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ ,
where
For $K_1$ , using the upper bound in the local limit theorem (cf. Theorem 5.8) and taking into account that $\phi \leqslant 1$ , we get
This and the fact that $\sqrt {{n}/{k}} \leqslant c$ imply
Using Lemma 4.6 and the fact that $m=[\varepsilon ^{1/2}n]$ , we get that uniformly in $z \in {\mathbb X}^+$ ,
We proceed to give an upper bound for $K_{2}$ , see (6.37). Recall that the function $(z,t) \mapsto \overline A_{m} (z,t)$ , which is involved in the definition of $K_2$ , is defined by (6.27) and does not, in general, belong to the space $\mathscr H^{\ \,+}_{\alpha }$ . We start by smoothing the indicator function in (6.27). Let $\kappa $ be a non-negative smooth compactly supported function in $[-1,1]$ such that $\int _{-1}^1 \kappa (u)\,du=1$ and set $\kappa _{\varepsilon }(u) = ({1}/{\varepsilon }) \kappa ({u}/{\varepsilon })$ for $u \in {\mathbb R}$ . Define
where $\chi _{\varepsilon }$ is the same as in (6.6) and $\overline \chi _{\varepsilon } = 1 - \chi _{\varepsilon }$ . Note that the function F is $\varepsilon /2$ -dominated by the function $G * \kappa _{\varepsilon /2}$ . By the identity
using the bounds (6.7) and $F(z, \cdot ) \leqslant G * \kappa _{\varepsilon /2} (z, \cdot )$ , we get that the function $\overline A_{m}$ is $\varepsilon /2$ -dominated by the function $\overline A_{m, \varepsilon }$ . Moreover, by Lemma 6.5, there exists a constant $c_{\varepsilon }$ such that for any $m \geqslant 1$ , the function $\overline A_{m, \varepsilon }$ belongs to $\mathscr H^{\ \,+}_{\alpha }$ and satisfies
Denote
Using the upper bound (6.1) and the fact that $\phi ^+ \leqslant 1$ , we obtain
For the first term on the right-hand side of (6.41), by the definition of $W_{m,\varepsilon }$ and Fubini’s theorem, we have
Using the duality (Lemma 2.12) yields that
As the measure $\nu $ is T-invariant, it follows that
where for the last line we made use of Lemma 2.2. By the Fuk inequality of Lemma 6.4 with $M=\varepsilon ^{-1/12}$ and $\varepsilon $ small enough, it follows that
Implementing this into (6.43), by (6.42), we have
where for the last line we made use of bounds similar to those in Lemma 6.5.
The norm $\| W_{m,\varepsilon } \|_{\mathscr H^{\ \,+}_{\alpha }}$ in the second term on the right-hand side of (6.41) is bounded using Lemma 6.5. Taking into account (6.40), we get
Therefore, from (6.41), (6.44) and (6.45), we derive the upper bound for $K_2$ : uniformly in $z \in {\mathbb X}^+$ and $t\leqslant t_0$ ,
Combining (6.28), (6.36), (6.37), (6.38) and (6.46), the lower bound (6.2) follows.
7 Proof of Theorem 1.7
As for Theorems 1.3 and 1.5, we first establish the result when g is in ${\mathscr B}^+$ . The general case of a function g in ${\mathscr B}$ will follow using the same method as in §4.
Theorem 7.1. Let $g \in {\mathscr B}^+$ be such that $\nu ^+ (g) = 0$ . Assume that for any $p \neq 0$ and $q \in {\mathbb R}$ , the function $p g + q$ is not cohomologous to a function with values in ${\mathbb Z}$ . Let F be a continuous compactly supported function on ${\mathbb X}^+ \times {\mathbb R}$ . Then, we have, uniformly in $z \in {\mathbb X}^+$ and t in a compact subset of ${\mathbb R}$ ,
In the proof of this theorem, we make use of several technical lemmas which are stated in the following. We say that a function G on ${\mathbb X}^+ \times {\mathbb R}$ is $\alpha $ -regular if there is a constant c such that for any $(z, t)$ and $(z', t')$ in ${\mathbb X}^+ \times {\mathbb R}$ , we have $|G(z,t) - G(z',t')| \leqslant c( |t-t'| + \alpha ^{\omega (z,z')} )$ . In other words, a function is $\alpha $ -regular if and only if it is Lipschitz continuous on ${\mathbb X}^+ \times {\mathbb R}$ when ${\mathbb R}$ is equipped with the standard distance and ${\mathbb X}^+$ is equipped with the distance $(z, z') \mapsto \alpha ^{\omega (z,z')}$ . The following result is similar to Lemma 6.5. It will allow us to smooth certain functions appearing in the proof of Theorem 7.1 in order to be able to apply Theorem 6.1. Recall that for $\varepsilon \in (0,1)$ , $\chi _{\varepsilon } (u) = 0$ for $u \leqslant -\varepsilon $ , $\chi _{\varepsilon } (u) = ({u+\varepsilon })/{\varepsilon }$ for $u \in (-\varepsilon ,0)$ , and $\chi _{\varepsilon } (u) = 1$ for $u \geqslant 0$ .
Lemma 7.2. Let $\alpha \in (0,1)$ and $g \in {\mathscr B}^+_{\alpha }$ be such that $\nu ^+ (g) = 0$ . Assume that g is not a coboundary. Let G be an $\alpha $ -regular function with compact support on ${\mathbb X}^+ \times {\mathbb R}$ . For $(z,t) \in {\mathbb X}^+ \times {\mathbb R}$ , $m \geqslant 1$ and $\varepsilon>0$ , define
Then $\overline \Psi _{m,\varepsilon } \in \mathscr H_{\alpha }^{\ \,+}$ and $\| \overline \Psi _{m,\varepsilon } \|_{\mathscr H_{\alpha }^{\ \,+}} \leqslant {c}/{\varepsilon \sqrt {m}}.$
Proof. It is enough to prove the lemma for a non-negative function G. Recall that
By Corollary 4.5, the first term is dominated by $c/\sqrt {m}$ for some constant $c>0$ .
For the second term, we start by noting that by Lemma 2.9, there exists a constant $c_0>0$ such that for any $z, z' \in {\mathbb X}^+$ with $z_0 = z_0'$ , $t \in {\mathbb R}$ and $y \in {\mathbb X}^-_{z}$ ,
where in the last inequality we used Corollary 2.10. It follows that
By using again Corollary 4.5, we get
In addition, as G is $\alpha $ -regular and has compact support, we have for any $z, z' \in {\mathbb X}^+$ with $z_0 = z_0'$ , and $t \in {\mathbb R}$ , by Lemma 2.9,
for some compactly supported continuous function H on ${\mathbb R}$ and some $c'>0$ . Again by Corollary 4.5, we get
Finally, for any $z, z' \in {\mathbb X}^+$ with $z_0 = z_0'$ , $t \in {\mathbb R}$ , we have
where $\theta $ is as in Lemma 2.3. By the Hölder continuous domination of $\theta $ in Lemma 2.3, we derive that
where the last inequality follows from Corollary 4.5. Putting together (7.1), (7.2) and (7.3) gives
The lemma follows.
Now we write a technical version of Theorem 7.1.
Lemma 7.3. Let $\alpha \in (0,1)$ and $g \in {\mathscr B}^+_{\alpha }$ be such that $\nu ^+ (g) = 0$ . Assume that for any $p \neq 0$ and $q \in {\mathbb R}$ , the function $p g + q$ is not cohomologous to a function with values in ${\mathbb Z}$ . Let $t \in {\mathbb R}$ . Then, for any $\varepsilon \in (0,\tfrac 18)$ and $z \in {\mathbb X}^+$ , and for any non-negative function F and non-negative $\alpha $ -regular compactly supported functions $G, H$ satisfying $H \leqslant _{\varepsilon } F \leqslant _{\varepsilon } G$ , we have
and
Proof. We first prove (7.4). As in (6.24), denote, for $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ ,
Set $m=[ n/2 ]$ and $k = n-m.$ By the Markov property we have that for any $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ ,
For any $z' \in {\mathbb X}^+$ and $t' \in {\mathbb R}$ , we set
By using $F \leqslant _{\varepsilon } G$ , we get that $\Psi _m \leqslant _{\varepsilon } \overline \Psi _m$ . Note that by Lemma 7.2, the function $\overline \Psi _m$ belongs to the space $\mathscr H_{\alpha }^{\ \,+}$ , so that we are exactly in the setting of Theorem 6.1. Therefore, using the bound (6.1) of Theorem 6.1, we get
For $J_1$ , applying the duality (Lemma 2.12), we deduce that
Using the conditioned central limit theorem (Theorem 1.5), we get
Therefore, we obtain
For $J_2$ , by Corollary 4.5, we have
Taking into account that $m=[n/2]$ and $k=n-m, $ we get $\limsup _{n \to \infty } n^{3/2} J_2 \leqslant c \varepsilon ^{1/4}.$
For $J_3$ , by Lemma 7.2, we have $\lim _{n \to \infty } n^{3/2} J_3 = 0$ . This finishes the proof of the upper bound. The proof of the lower bound can be carried out in the same way.
From Lemma 7.3, we get Theorem 7.1 by a standard approximation procedure.
Lemma 7.4. Fix $\alpha \in (0,1)$ . Let F be a non-negative continuous compactly supported function on ${\mathbb X}^+ \times {\mathbb R}$ . Then, there exist a decreasing sequence $(G_k)_{k \geqslant 1}$ and an increasing sequence $(H_k)_{k \geqslant 1}$ of compactly supported $\alpha $ -regular functions, such that $H_k \leqslant _{1/k} F \leqslant _{1/k} G_k$ for any $k \geqslant 1$ , and $G_k$ and $H_k$ converge uniformly to F as $k \to \infty $ .
From Theorem 7.1 we deduce a new lemma in which the target function F may depend on the past coordinates.
Lemma 7.5. Let $g \in {\mathscr B}^+$ be such that $\nu ^+ (g) = 0$ . Assume that for any $p \neq 0$ and ${q \in {\mathbb R}}$ , the function $p g + q$ is not cohomologous to a function with values in ${\mathbb Z}$ . Let F be a continuous compactly supported function on ${\mathbb X} \times {\mathbb R}$ . Then, uniformly in $z \in {\mathbb X}_+$ and t in a compact subset of ${\mathbb R}$ ,
Proof. As in the proof of Lemma 4.7, it suffices to prove this result when F is of the form
, where $a \in A^{\{-m,\ldots ,-1\}}$ satisfies $M(a_{i-1}, a_{i}) = 1$ for $-m+1 \leqslant i \leqslant -1$ , and $F_1$ is a continuous compactly supported function on ${\mathbb X}^+\times {\mathbb R}$ . For such a function, we have
where, for $(z', t') \in {\mathbb X}^+ \times {\mathbb R}$ ,
As $F_2(\cdot , t')$ depends only on the future, we can apply Theorem 7.1, which gives
To conclude, it remains to show that
Indeed, by the definition of the measure $\mu ^{(-g)}$ (see Theorem 1.1) and by using Lemma 2.2, we get
which ends the proof of the lemma.
Now we place a target on the starting point $y \in {\mathbb X}^-_z$ .
Lemma 7.6. Let $g \in {\mathscr B}^+$ be such that $\nu ^+ (g) = 0$ . Assume that for any $p \neq 0$ and $q \in {\mathbb R}$ , the function $p g + q$ is not cohomologous to a function with values in ${\mathbb Z}$ . Then, for any $(z, t) \in {\mathbb X}^+ \times {\mathbb R}$ and any continuous compactly supported function F on ${\mathbb X}_z^- \times {\mathbb X} \times {\mathbb R}$ , we have
Proof. As usual, it suffices to prove the lemma when F is of the form , where $a \in A_z^m$ and G is a continuous compactly supported function on ${\mathbb X} \times {\mathbb R}$ .
If $t + S_k g(T^{m-k} (a \cdot z)) \geqslant 0$ for every $1 \leqslant k \leqslant m$ , we have that for $n> m$ ,
By Lemma 7.5, as $n \to \infty $ , the latter quantity converges to
which, by the definition of measure $\check {\mu }^{g,-}_{z,t}$ (see (3.21)), is equal to
If there exists $1 \leqslant k \leqslant m$ with $t + S_k g(T^{m-k} (a \cdot z)) <0$ , we have $\check {\mu }^{g,-}_{z,t} (\mathbb C_{a,z}) = 0$ and
for $n> k$ . The conclusion follows.
As usual, from Lemma 7.6, we want to deduce the analogous result for functions which depend only on finitely many negative coordinates. We use the following easy formula that relates the measures $\mu ^g$ and $\mu ^{g \circ T}\!.$
Lemma 7.7. Let $g \in {\mathscr B}$ be such that $\nu (g) = 0$ and g is not a coboundary. Then, for any continuous compactly supported function F on ${\mathbb X} \times {\mathbb R}$ , we have
Proof. By using the relation $\tau _t^{g \circ T} = \tau _t^{g} \circ T$ , we get
as desired.
Lemma 7.8. Let $g \in {\mathscr B}$ be such that $\nu (g) = 0$ and there exists $m \geqslant 0$ with $g \circ T^m \in {\mathscr B}^+$ . Assume that for any $p \neq 0$ and $q \in {\mathbb R}$ , the function $p g + q$ is not cohomologous to a function with values in ${\mathbb Z}$ . Then, for any $(z, t) \in {\mathbb X}^+ \times {\mathbb R}$ and any continuous compactly supported function F on ${\mathbb X}_z^- \times {\mathbb X} \times {\mathbb R}$ , we have
Proof. As in Lemma 3.15, for $a \in A_z^m$ , set $F_a$ to be the function on ${\mathbb X}^-_{a \cdot z} \times {\mathbb X} \times {\mathbb R}$ defined by $F_a (y, x, t) = F (y \cdot a, T^m x, t)$ . Then we have, by setting $h = g \circ T^m$ ,
By Lemma 7.6, as $n \to \infty $ , this converges to
By (3.24), the latter quantity is equal to
As $h = g \circ T^m$ , the conclusion now follows from Lemma 7.7.
Now we can give a result for any function g in ${\mathscr B}$ .
Lemma 7.9. Let $g \in {\mathscr B}$ be such that $\nu (g) = 0$ . Assume that for any $p \neq 0$ and $q \in {\mathbb R}$ , the function $p g + q$ is not cohomologous to a function with values in ${\mathbb Z}$ . Then, for any continuous compactly supported function F on ${\mathbb X} \times {\mathbb X} \times {\mathbb R} \times {\mathbb R}$ , we have
Proof. We can assume that the function F is non-negative. For $(z,t) \in {\mathbb X}^+ \times {\mathbb R}$ , denote
Let $(g_m)_{m \geqslant 0}$ be the sequence of Hölder continuous functions as in Lemma 2.11. For any $n, m \geqslant 0$ , we set
and
For $z \in {\mathbb X}^+$ and $t \in {\mathbb R}$ , it holds that
By taking the limit as $n \to \infty $ , we get by Lemma 7.8,
On the one hand, after integrating over ${\mathbb X}^+ \times {\mathbb R}$ in (7.7) with respect to the product of $\nu ^+$ with the Lebesgue measure, by Fatou’s lemma and Lemma 7.8, we get
To conclude, we need to show the reverse Fatou property holds. To this aim, we choose a non-negative continuous compactly supported function G on ${\mathbb R}$ such that for any $(x, x', t, t') \in {\mathbb X} \times {\mathbb X} \times {\mathbb R} \times {\mathbb R}$ , one has $ F_0^+(x, x', t, t') \leqslant G(t) G(t'). $ Then, we get for $(z,t) \in {\mathbb X}^+ \times {\mathbb R}$ ,
By Lemma 7.1, $U_n(z,t)$ converges uniformly in $(z,t) \in {\mathbb X}^+ \times {\mathbb R}$ . Therefore, by applying Fatou’s lemma to the sequence $U_n(z,t) - W_{n} (z,t)$ , we get by integrating over ${\mathbb X}^+ \times {\mathbb R}$ in (7.7) with respect to the product of $\nu ^+$ with the Lebesgue measure,
By letting $m \to \infty $ , the conclusion follows from (7.8), (7.9) and Lemma 3.19.
Acknowledgements
The authors would like to express their gratitude to the referee for the very careful reading of the article and for the valuable remarks that have contributed to improving the presentation. Ion Grama and Hui Xiao are supported by DFG grant ME 4473/2-1. Hui Xiao is also supported by the National Natural Science Foundation of China (Grant No. 12288201).