We prove a new large deviation inequality with applicationswhen projecting a density on a wavelet basis.

The statistical properties of the likelihood ratio test statistic (LRTS) for autoregressive regime-switching models are addressed in this paper. This question is particularly important for estimating the number of regimes in the model. Our purpose is to extend the existing results for mixtures [X. Liu and Y. Shao, Ann. Stat. 31 (2003) 807–832] and hidden Markov chains [E. Gassiat, Ann. Inst. Henri Poincaré 38 (2002) 897–906]. First, we study the case of mixtures of autoregressive models (i.e. independent regime switches). In this framework, we give sufficient conditions to keep the LRTS tight and compute its the asymptotic distribution. Second, we consider the extension of the ideas in Gassiat [Ann. Inst. Henri Poincaré 38 (2002) 897–906] to autoregressive models with regimes switches according to a Markov chain. In this case, it is shown that the marginal likelihood is no longer a contrast function and cannot be used to select the number of regimes. Some numerical examples illustrate the results and their convergence properties.

In this paper, we investigate a nonparametric approach to provide a recursive estimatorof the transition density of a piecewise-deterministic Markov process, from only oneobservation of the path within a long time. In this framework, we do not observe a Markovchain with transition kernel of interest. Fortunately, one may write the transitiondensity of interest as the ratio of the invariant distributions of two embedded chains ofthe process. Our method consists in estimating these invariant measures. We state a resultof consistency and a central limit theorem under some general assumptions about the mainfeatures of the process. A simulation study illustrates the well asymptotic behavior ofour estimator.

We investigate the dissipativity properties of a class of scalar secondorder parabolic partial differential equations with time-dependentcoefficients. We provide explicit condition on the drift term which ensurethat the relative entropy of one particular orbit with respect to some otherone decreases to zero. The decay rate is obtained explicitly by the use ofa Sobolev logarithmic inequality for the associated semigroup, which is derived by an adaptation of Bakry's Γ-calculus.As a byproduct, the systematic method for constructing entropieswhich we propose here also yields the well-known intermediate asymptotics for the heat equation in a very quick way, and without having to rescale the original equation.

A branching random motion on a line, with abrupt changes of direction, is studied. The branching mechanism, being independent of random motion, and intensities of reverses are defined by a particle's current direction. A solution of a certain hyperbolic system of coupled non-linear equations (Kolmogorov type backward equation) has a so-called McKean representation via such processes. Commonly this system possesses travelling-wave solutions. The convergence of solutions with Heaviside terminal data to the travelling waves is discussed. The paper realizes the McKean's program for the Kolmogorov-Petrovskii-Piskunov equation in this case. The Feynman-Kac formula plays a key role.

We consider the standard first passage percolation model in ℤd for d ≥ 2 and we study the maximal flow from the upper half part to the lower half part (respectively from the top to the bottom) of a cylinder whose basis is a hyperrectangle of sidelength proportional to n and whose height is h(n) for a certain height function h. We denote this maximal flow by τn (respectively φn). We emphasize the fact that the cylinder may be tilted. We look at the probability that these flows, rescaled by the surface of the basis of the cylinder, are greater than ν(v) + ε for some positive ε, where ν(v) is the almost sure limit of the rescaled variable τn when n goes to infinity. On one hand, we prove that the speed of decay of this probability in the case of the variable τn depends on the tail of the distribution of the capacities of the edges: it can decay exponentially fast with nd−1, or with nd−1min(n,h(n)), or at an intermediate regime. On the other hand, we prove that this probability in the case of the variable φn decays exponentially fast with the volume of the cylinder as soon as the law of the capacity of the edges admits one exponential moment; the importance of this result is however limited by the fact that ν(v) is not in general the almost sure limit of the rescaled maximal flow φn, but it is the case at least when the height h(n) of the cylinder is negligible compared to n.

We continue the investigation started in a previous paper, onweak convergence to infinitely divisible distributions with finitevariance. In the present paper, we study this problem for someweakly dependent random variables, including in particularassociated sequences. We obtain minimal conditions expressed interms of individual random variables. As in the i.i.d. case, wedescribe the convergence to the Gaussian and the purelynon-Gaussian parts of the infinitely divisible limit. We alsodiscuss the rate of Poisson convergence and emphasize the specialcase of Bernoulli random variables. The proofs aremainly based on Lindeberg's method.

In many applications, we assume that two random observations x andy are generated according to independent Poisson distributions\hbox{$\PPP(\lambdaS)$}𝒫(λS)and \hbox{$\PPP(\muT)$}𝒫(μT)and we are interested in performing statistical inference on the ratioφ = λ / μ of the twoincidence rates. In vaccine efficacy trials, x and y aretypically the numbers of cases in the vaccine and the control groups respectively,φ is called the relative risk and the statistical model is called‘partial immunity model’. In this paper we start by defining a natural semi-conjugatefamily of prior distributions for this model, allowing straightforward computation of theposterior inference. Following theory on reference priors, we define the reference priorfor the partial immunity model when φ is the parameter of interest. Wealso define a family of reference priors with partial information on μwhile remaining uninformative about φ. We notice that these priors belongto the semi-conjugate family. We then demonstrate using numerical examples that Bayesiancredible intervals for φ enjoy attractive frequentist properties whenusing reference priors, a typical property of reference priors.

We have seen in a previous article how the theory of “rough paths”allows us to construct solutions of differential equations driven by processes generated by divergence form operators. In this article, we study a convergence criterion which implies that one can interchange the integral with the limit of a family of stochastic processes generated by divergence form operators. As a corollary, we identify stochastic integrals constructed with the theory of rough paths with Stratonovich or Itô integrals already constructedfor stochastic processes generated by divergence form operators by using time-reversal techniques.

We study the convergence rate of randomly truncated stochastic algorithms, which consistin the truncation of the standard Robbins–Monro procedure on an increasing sequence ofcompact sets. Such a truncation is often required in practice to ensure convergence whenstandard algorithms fail because the expected-value function grows too fast. In this work,we give a self contained proof of a central limit theorem for this algorithm under localassumptions on the expected-value function, which are fairly easy to check in practice.

We prove that the large deviation principle holds for a class ofprocesses inspired by semi-Markov additive processes. For theprocesses we consider, the sojourn times in the phase process neednot be independent and identically distributed. Moreover thestate selection process need not be independent of the sojourntimes. We assume that the phase process takes values in a finite set andthat the order in which elements in the set, called states, arevisited is selected stochastically. The sojourn times determinehow long the phase process spends in a state once it has beenselected. The main tool is a representation formula for the samplepaths of the empirical laws of the phase process. Then, based onassumed joint large deviation behavior of the state selection andsojourn processes, we prove that the empirical laws of the phaseprocess satisfy a sample path large deviation principle. From thislarge deviation principle, the large deviations behavior of aclass of modulated additive processes is deduced. As an illustration of the utility of the general results, we providean alternate proof of results for modulated Lévy processes. As apractical application of the results, we calculate the large deviationrate function for a processes that arises as the InternationalTelecommunications Union's standardized stochastic model of two-wayconversational speech.

This paper introduces a computationally tractable density estimatorthat has the same asymptotic variance as the classical Nadaraya-Watsondensity estimator but whose asymptotic bias is zero. We achieve this resultusing a two stage estimator that applies a multiplicative bias correctionto an oversmooth pilot estimator. Simulations show that our asymptotic results are available for samples as low as n = 50, where we see an improvement of as much as 20% over the traditionnal estimator.

We investigate the small deviations under various norms for stable processes defined by the convolution of a smooth function $f : \; ]0, +\infty[ \;\to \mathbb{R}$ with a real SαS Lévy process. We show that the small ball exponent is uniquely determined by the norm and by the behaviour of f at zero, which extends the results of Lifshits and Simon, Ann. Inst. H. Poincaré Probab. Statist.41 (2005) 725–752where this was proved for f being a power function (Riemann-Liouville processes). In the Gaussian case, the same generality as Lifshits and Simon, Ann. Inst. H. Poincaré Probab. Statist.41 (2005) 725–752 is obtained with respect to the norms, thanks to a weak decorrelation inequality due to Li, Elec. Comm. Probab.4 (1999) 111–118. In the more difficult non-Gaussian case, we use a different method relying on comparison of entropy numbers and restrict ourselves to Hölder and L p -norms.

In the problem of signal detectionin Gaussian white noisewe show asymptotic minimaxity of kernel-based tests. The test statisticsequal L 2-norms of kernel estimates.The sets of alternatives are essentially nonparametric and are defined asthe sets of all signals such that the L 2-norms of signal smoothed by the kernels exceed some constants pε > 0.The constant pε depends on the power ϵof noise and pε → 0 as ε → 0.Similar statements are proved also if an additional informationon a signal smoothness is given.By theorems on asymptotic equivalence of statistical experimentsthese results are extended to the problems of testing nonparametrichypotheseson density and regression. The exact asymptotically minimaxlower bounds of type II error probabilities are pointed out forall these settings. Similar results are also obtained for the problemsof testing parametric hypotheses versus nonparametric sets of alternatives.

Let F n be the empirical distribution function (df) pertainingto independent random variables with continuous df F. Weinvestigate the minimizing point $\hat\tau_n$ of the empiricalprocess Fn - F0 , where F 0 is another df which differs fromF. If F and F 0 are locally Hölder-continuous of orderα at a point τ our main result states that $n^{1/\alpha}(\hat\tau_n - \tau)$ converges in distribution. Thelimit variable is the almost sure unique minimizing point of atwo-sided time-transformed homogeneous Poisson-process with adrift. The time-transformation and the drift-function are of thetype |t|α.

The paper develops an approach to optimal design problems based on application of abstract optimisation principles in the space of measures. Various design criteria and constraints, such as bounded density, fixed barycentre, fixed variance, etc. are treated in a unified manner providing a universal variant of the Kiefer-Wolfowitz theorem and giving a full spectrum of optimality criteria for particular cases. Incorporating the optimal design problems into conventional optimisation framework makes it possible to use the whole arsenal of descent algorithms from the general optimisation literature for finding optimal designs. The corresponding steepest descent involves adding a signed measure at every step and converges faster than the conventional sequential algorithms used to construct optimal designs. We study a new class of design problems when the observation points are distributed according to a Poisson point process arising in the situation when the total control on the placement of measurements is impossible.

Let X1,...,Xn1 be arandom sample from a population with mean µ 1 and variance $\sigma_1^2$ , and X1,...,Xn1 be a random sample fromanother population with mean µ 2 and variance $\sigma_2^2$ independent of{Xi,1 ≤ i ≤ n1 }.Consider the twosample t-statistic $ T={{\bar X-\bar Y-(\mu_1-\mu_2)} \over\sqrt{s_1^2/n_1+s_2^2/n_2}}$ . This paper shows thatln P(T ≥ x) ~ -x²/2 for any x := x(n1,n2) satisfying x → ∞, x = o(n1 + n2)1/2 as n1,n2 → ∞ provided 0 < c1 ≤ n1/n2 ≤ c2 < ∞. If, inaddition, E|X1|3 < ∞, E|Y1|3 < ∞, then $\frac{P(T \geq x)}{1-\Phi(x)} \to 1 $ holds uniformly in x ∈ (O,o((n1 + n2)1/6))

We consider a system of fully coupled forward-backward stochasticdifferential equations. First we generalize the results ofPardoux-Tang [7] concerning the regularity of the solutions withrespect to initial conditions. Then, we prove that in some particularcases this system leads to aprobabilistic representation of solutions of a second-order PDE whosesecond order coefficients depend on the gradient of the solution. Wethen give some examples in dimension 1 and dimension 2 for which theassumptions are easy to check.

Sur un ensemble fini, on s'intéresse aux minorations linéaires du trou spectral d'un noyau markovien réversible, en terme de la constante isopérimétrique associée. On montre que la constanteoptimale est l'inverse du cardinal de l'ensemble moins un, mais on verra aussi comment il est possible de l'améliorer dans certaines situations particulières (arbres pointés radiaux à nombre fini de générations). Une application des inégalités précédentes est de retrouver immédiatement le comportement à basse température des trous spectraux associés aux algorithmes de recuit simulé généralisés finis. Une approche similaire permet également d'obtenir très simplement une caractérisation de l'énergie virtuelle due à Trouvé.

Soit U une fonction définie sur un ensemble fini E muni d'unnoyau markovien irréductible M. L'objectif du papier est de comparerthéoriquement deux procédures stochastiques de minimisation globale de U :le recuit simulé et un algorithme génétique.Pour ceci on se placera dans la situation idéalisée d'une infinité de particules disponibleset nous feronsune hypothèse commode d'existence de suffisamment de symétries du cadre (E,M,U).On verra notamment que contrairement au recuit simulé, toute évolution logarithmiquede l'inverse de la température conduit asymptotiquement à concentrer les dynamiques de Feynman-Kac recuitessurcertains minima globaux de U (du moins si M ne fait pas sortir de leur ensemble en un temps presque immédiat).Par contre, ces dernières ont tendance à oublier plus difficilement leur condition initiale et on quantifiera précisément cette propriété. Enfin on présentera les conjectures correspondantes dans une situation plus générale.