Consider an irreducible, aperiodic and positive recurrent discrete time Markov chain(Xn,n ≥0) with invariant distribution μ. We shall investigate thelong time behaviour of some functionals of the chain, in particular the additivefunctional \hbox{$S_{n}=\sum_{i=1}^{n}f(X_{i})$}${\mathit{S}}_{\mathit{n}}\mathrm{=}{\sum }_{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{n}}\mathit{f}\mathrm{\left(}{\mathit{X}}_{\mathit{i}}\mathrm{\right)}$ for a possibly non square integrable functionf. To thisend we shall link ergodic properties of the chain to mixing properties, extending knownresults in the continuous time case. We will then use existing results of convergence tostable distributions, obtained in [M. Denker and A. Jakubowski, Stat. Probab.Lett. 8 (1989) 477–483; M. Tyran-Kaminska, StochasticProcess. Appl. 120 (2010) 1629–1650; D. Krizmanic, Ph.D. thesis(2010); B. Basrak, D. Krizmanic and J. Segers, Ann. Probab. 40(2012) 2008–2033] for stationary mixing sequences. Contrary to the usualL^2$\mathrm{L}{\mathrm{2}}_{}^{}$ frameworkstudied in [P. Cattiaux, D. Chafai and A. Guillin, ALEA, Lat. Am. J. Probab. Math.Stat. 9 (2012) 337–382], where weak forms of ergodicity aresufficient to ensure the validity of the Central Limit Theorem, we will need here strongergodic properties: the existence of a spectral gap, hyperboundedness (orhypercontractivity). These properties are also discussed. Finally we give explicitexamples.

This paper presents a practical and simple fully nonparametric multivariate smoothingprocedure that adapts to the underlying smoothness of the true regression function. Ourestimator is easily computed by successive application of existing base smoothers (withoutthe need of selecting an optimal smoothing parameter), such as thin-plate spline or kernelsmoothers. The resulting smoother has better out of sample predictive capabilities thanthe underlying base smoother, or competing structurally constrained models (MARS, GAM) forsmall dimension (3 ≤ d ≤7) and moderate sample size n ≤ 1000. Moreover our estimator is still usefulwhen d > 10and to our knowledge, no other adaptive fully nonparametric regression estimator isavailable without constrained assumption such as additivity for example. On a realexample, the Boston Housing Data, our method reduces the out of sample prediction error by20%. An R package ibr, available at CRAN, implements the proposedmultivariate nonparametric method in R.

With direct and simple proofs, we establish Poincaré type inequalities (includingPoincaré inequalities, weak Poincaré inequalities and super Poincaré inequalities),entropy inequalities and Beckner-type inequalities for non-local Dirichlet forms. Theproofs are efficient for non-local Dirichlet forms with general jump kernel, and also workfor Lp(p>1) settings. Our results yield a new sufficient condition forfractional Poincaré inequalities, which were recently studied in [P.T. Gressman,J. Funct. Anal. 265 (2013) 867–889. C. Mouhot, E. Russ andY. Sire, J. Math. Pures Appl. 95 (2011) 72–84.] To ourknowledge this is the first result providing entropy inequalities and Beckner-typeinequalities for measures more general than Lévy measures.

We use multivariate total positivity theory to exhibit new families of peacocks. As theauthors of [F. Hirsch, C. Profeta, B. Roynette and M. Yor, Peacocks and associatedmartingales vol. 3. Bocconi-Springer (2011)], our guiding example is the resultof Carr−Ewald−Xiao [P. Carr, C.-O. Ewald and Y. Xiao,Finance Res. Lett. 5 (2008) 162–171]. We shall introducethe notion of strong conditional monotonicity. This concept is strictly more restrictivethan the conditional monotonicity as defined in [F. Hirsch, C. Profeta, B. Roynette and M.Yor, Peacocks and associated martingales, vol. 3. Bocconi-Springer (2011)] (see also [R.H.Berk, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 42(1978) 303–307], [A.M. Bogso, C. Profeta and B. Roynette, Lect. Notes Math.Springer, Berlin (2012) 281–315.] and [M. Shaked and J.G. Shanthikumar,Probab. Math. Statistics. Academic Press, Boston (1994)].). There aremany random vectors which are strongly conditionally monotone (SCM). Indeed, we shallprove that multivariate totally positive of order 2 (MTP2) random vectors are SCM. As aconsequence, stochastic processes with MTP2 finite-dimensional marginals are SCM. This familyincludes processes with independent and log-concave increments, and one-dimensionaldiffusions which have absolutely continuous transition kernels.

In [A. Genadot and M. Thieullen, Averaging for a fully coupled piecewise-deterministicmarkov process in infinite dimensions. Adv. Appl. Probab. 44(2012) 749–773], the authors addressed the question of averaging for a slow-fastPiecewise Deterministic Markov Process (PDMP) in infinite dimensions. In the presentpaper, we carry on and complete this work by the mathematical analysis of the fluctuationsof the slow-fast system around the averaged limit. A central limit theorem is derived andthe associated Langevin approximation is considered. The motivation for this work is thestudy of stochastic conductance based neuron models which describe the propagation of anaction potential along a nerve fiber.

We consider dependence coefficients for stationary Markov chains. We emphasize on someequivalencies for reversible Markov chains. We improve some known results and provide anecessary condition for Markov chains based on Archimedean copulas to be exponentialρ-mixing.We analyse the example of the Mardia and Frechet copula families using small sets.

In a multiple testing context, we consider a semiparametric mixture model with two components where one component is known and corresponds to the distribution of p-values under the null hypothesis and the other component f is nonparametric and stands for the distribution under the alternative hypothesis. Motivated by the issue of local false discovery rate estimation, we focus here on the estimation of the nonparametric unknown component f in the mixture, relying on a preliminary estimator of the unknown proportion θ of true null hypotheses. We propose and study the asymptotic properties of two different estimators for this unknown component. The first estimator is a randomly weighted kernel estimator. We establish an upper bound for its pointwise quadratic risk, exhibiting the classical nonparametric rate of convergence over a class of Hölder densities. To our knowledge, this is the first result establishing convergence as well as corresponding rate for the estimation of the unknown component in this nonparametric mixture. The second estimator is a maximum smoothed likelihood estimator. It is computed through an iterative algorithm, for which we establish a descent property. In addition, these estimators are used in a multiple testing procedure in order to estimate the local false discovery rate. Their respective performances are then compared on synthetic data.

We study the numerical approximation of doubly reflected backward stochastic differentialequations with intermittent upper barrier (RIBSDEs). These denote reflected BSDEs in whichthe upper barrier is only active on certain random time intervals. From the point of viewof financial interpretation, RIBSDEs arise as pricing equations of game options withconstrained callability. In a Markovian set-up we prove a convergence rate for atime-discretization scheme by simulation to an RIBSDE. We also characterize the solutionof an RIBSDE as the largest viscosity subsolution of a related system of variationalinequalities, and we establish the convergence of a deterministic numerical scheme forthat problem. Due to the potentially very high dimension of the system of variationalinequalities, this approach is not always practical. We thus subsequently prove aconvergence rate for a time-discretisation scheme by simulation to an RIBSDE.

We consider the high order moments estimator of the frontier of a random pair, introduced by [S. Girard, A. Guillou and G. Stupfler, J. Multivariate Anal. 116 (2013) 172–189]. In the present paper, we show that this estimator is strongly uniformly consistent on compact sets and its rate of convergence is given when the conditional cumulative distribution function belongs to the Hall class of distribution functions.

Pareto distributions are most popular for modeling heavy tailed data. Here, we obtain weak limits of a sequence of extremal and a sequence of additive processes constructed by a series of Bernoulli point processes with bivariate Pareto space components. For the limiting processes we derive the one dimensional distributions in explicit forms. Some of the main properties of these distributions are also proved.

In this note we propose an exact simulation algorithm for the solution of (1)

\begin{equation} \label{eds-intro} {\rm d}X_t={\rm d}W_t+\bar{b}(X_t){\rm d}t,\quad X_0=x, \end{equation}$\mathrm{d}{\mathit{X}}_{\mathit{t}}\mathrm{=}\mathrm{d}{\mathit{W}}_{\mathit{t}}\mathrm{+}\mathit{b̅}\mathrm{\left(}{\mathit{X}}_{\mathit{t}}\mathrm{\right)}\mathrm{d}\mathit{t,}\hspace{1em}{\mathit{X}}_{\mathrm{0}}\mathrm{=}\mathit{x,}$

\hbox{$\bar{b}$}b̅

\hbox{$\bar{b}(0+)\neq \bar{b}(0-)$}b̅(0 + ) ≠b̅(0 −)

(2)

\begin{equation} \label{edsbeta} {\rm d}X^\beta_t={\rm d}W_t+\bar{b}(X^\beta_t){\rm d}t + \beta {\rm d}L^0_t(X^\beta),\quad X_0=x \end{equation}$\mathrm{d}{\mathit{X}}_{\mathit{t}}^{\mathit{\beta }}\mathrm{=}\mathrm{d}{\mathit{W}}_{\mathit{t}}\mathrm{+}\mathit{b̅}\mathrm{\left(}{\mathit{X}}_{\mathit{t}}^{\mathit{\beta }}\mathrm{\right)}\mathrm{d}\mathit{t}\mathrm{+}\mathit{\beta }\mathrm{d}{\mathit{L}}_{\mathit{t}}^{\mathrm{0}}\mathrm{\left(}{\mathit{X}}^{\mathit{\beta }}\mathrm{\right)}\mathit{,}\hspace{1em}{\mathit{X}}_{\mathrm{0}}\mathrm{=}\mathit{x}$

We consider probability measures supported on a finite discrete interval [0, n]. We introduce a newfinite difference operator ∇n, defined as a linear combination ofleft and right finite differences. We show that this operator ∇n plays a keyrole in a new Poincaré (spectral gap) inequality with respect to binomial weights, withthe orthogonal Krawtchouk polynomials acting as eigenfunctions of the relevant operator.We briefly discuss the relationship of this operator to the problem of optimal transportof probability measures.

We study asymptotic behavior of Markov chain Monte Carlo (MCMC) procedures. Sometimes theperformances of MCMC procedures are poor and there are great importance for the study ofsuch behavior. In this paper we call degeneracy for a particular type of poorperformances. We show some equivalent conditions for degeneracy. As an application, weconsider the cumulative probit model. It is well known that the natural data augmentation(DA) procedure does not work well for this model and the so-called parameter-expanded dataaugmentation (PX-DA) procedure is considered to be a remedy for it. In the sense ofdegeneracy, the PX-DA procedure is better than the DA procedure. However, when the numberof categories is large, both procedures are degenerate and so the PX-DA procedure may notprovide good estimate for the posterior distribution.

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 consider the segmentation problem of Poisson and negative binomial (i.e.overdispersed Poisson) rate distributions. In segmentation, an important issueremains the choice of the number of segments. To this end, we propose a penalized-likelihood estimator where the penaltyfunction is constructed in a non-asymptotic context following the works of L. Birgé and P.Massart. The resulting estimator is proved to satisfy an oracle inequality. Theperformances of our criterion is assessed using simulated and real datasets in the RNA-seqdata analysis context.

This paper deals with variable selection in regression and binary classificationframeworks. It proposes an automatic and exhaustive procedure which relies on the use ofthe CART algorithm and on model selection via penalization. This work, of theoreticalnature, aims at determining adequate penalties, i.e. penalties whichallow achievement of oracle type inequalities justifying the performance of the proposedprocedure. Since the exhaustive procedure cannot be realized when the number of variablesis too large, a more practical procedure is also proposed and still theoreticallyvalidated. A simulation study completes the theoretical results.

Consider a mean-reverting equation, generalized in the sense it is driven by a1-dimensional centeredGaussian process with Hölder continuous paths on [0,T] (T> 0). Taking thatequation in rough paths sense only gives local existence of the solution because thenon-explosion condition is not satisfied in general. Under natural assumptions, by usingspecific methods, we show the global existence and uniqueness of the solution, itsintegrability, the continuity and differentiability of the associated Itô map, and weprovide an Lp-convergingapproximation with a rate of convergence (p ≫ 1). The regularity of the Itô map ensures a largedeviation principle, and the existence of a density with respect to Lebesgue’s measure,for the solution of that generalized mean-reverting equation. Finally, we study ageneralized mean-reverting pharmacokinetic model.

Let \hbox{$\mathbb{F}^{}_{}$}$F_{}^{}$ be a filtration and τ be a random time. Let \hbox{$\mathbb{G}^{}_{}$}$G_{}^{}$ be the progressive enlargement of \hbox{$\mathbb{F}^{}_{}$}$F_{}^{}$ with τ. We study the following formula, called the optional splitting formula: For any \hbox{$\mathbb{G}^{}_{}$}$G_{}^{}$-optional process Y, there exists an \hbox{$\mathbb{F}^{}_{}$}$F_{}^{}$-optional process Y′ and a function Y′′ defined on [0,∞] × (ℝ+ × Ω) being \hbox{$\mathcal{B}[0,\infty]\otimes\mathcal{O}(\mathbb{F}^{}_{})$}$\mathrm{ℬ}\mathrm{\left[}\mathrm{0}\mathit{,}\mathrm{\infty }\mathrm{\right]}\mathrm{\otimes }\mathrm{𝒪}\mathrm{\left(}{F}_{}^{}\mathrm{\right)}$ measurable, such that \hbox{$Y=Y'\mathds{1}^{}_{[0,\tau)}+Y''(\tau)\mathbb{1}^{}_{[\tau,\infty)}.$}$\mathit{Y}\mathrm{=}{\mathit{Y}}^{\mathrm{\prime }}{1}_{\mathrm{\left[}\mathrm{0}\mathit{,\tau }\mathrm{\right)}}^{}\mathrm{+}{\mathit{Y}}^{\mathrm{\prime \prime }}\mathrm{\left(}\mathit{\tau }\mathrm{\right)}{1}_{\mathrm{\left[}\mathit{\tau ,}\mathrm{\infty }\mathrm{\right)}}^{}\mathit{.}$ (This formula can also be formulated for multiple random times τ1,...,τk). We are interested in this formula because of its fundamental role in many recent papers on credit risk modeling, and also because of the fact that its validity is limited in scope and this limitation is not sufficiently underlined. In this paper we will determine the circumstances in which the optional splitting formula is valid. We will then develop practical sufficient conditions for that validity. Incidentally, our results reveal a close relationship between the optional splitting formula and several measurability questions encountered in credit risk modeling. That relationship allows us to provide simple answers to these questions.

The Gauss−Minkowskicorrespondence in ℝ2 states the existence of a homeomorphism between theprobability measures μ on [0,2π] such that \hbox{$\int_0^{2\pi} {\rm e}^{ix}{\rm d}\mu(x)=0$}${{}^{\mathrm{\int }}}_{\mathrm{0}}^{\mathrm{2}\mathit{\pi }}{\mathrm{e}}^{\mathit{ix}}\mathrm{d}\mathit{\mu }\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\mathrm{0}$ and the compact convex sets (CCS) of the plane withperimeter 1. In this article, we bring out explicit formulas relating the border of a CCSto its probability measure. As a consequence, we show that some natural operations on CCS– for example, the Minkowski sum – have natural translations in terms of probabilitymeasure operations, and reciprocally, the convolution of measures translates into a newnotion of convolution of CCS. Additionally, we give a proof that a polygonal curveassociated with a sample of n random variables (satisfying \hbox{$\int_0^{2\pi} {\rm e}^{ix}{\rm d}\mu(x)=0$}${{}^{\mathrm{\int }}}_{\mathrm{0}}^{\mathrm{2}\mathit{\pi }}{\mathrm{e}}^{\mathit{ix}}\mathrm{d}\mathit{\mu }\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\mathrm{0}$) converges to a CCS associated with μ at speed √n, a result much similar to the convergence of theempirical process in statistics. Finally, we employ this correspondence to present modelsof smooth random CCS and simulations.

We study the estimation of the mean function of a continuous-time stochastic process andits derivatives. The covariance function of the process is assumed to be nonparametric andto satisfy mild smoothness conditions. Assuming that n independent realizationsof the process are observed at a sampling design of size N generated by a positivedensity, we derive the asymptotic bias and variance of the local polynomial estimator asn,Nincrease to infinity. We deduce optimal sampling densities, optimal bandwidths, andpropose a new plug-in bandwidth selection method. We establish the asymptotic performanceof the plug-in bandwidth estimator and we compare, in a simulation study, its performancefor finite sizes n,N to the cross-validation and the optimalbandwidths. A software implementation of the plug-in method is available in the Renvironment.