Consider an irreducible, aperiodic and positive recurrent discrete time Markov chain (Xn,n ≥ 0) with invariant distribution μ. We shall investigate the long time behaviour of some functionals of the chain, in particular the additive functional \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 function f. To this end we shall link ergodic properties of the chain to mixing properties, extending known results in the continuous time case. We will then use existing results of convergence to stable distributions, obtained in [M. Denker and A. Jakubowski, Stat. Probab. Lett. 8 (1989) 477–483; M. Tyran-Kaminska, Stochastic Process. 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 usual L^2$\mathrm{L}{\mathrm{2}}_{}^{}$ framework studied in [P. Cattiaux, D. Chafai and A. Guillin, ALEA, Lat. Am. J. Probab. Math. Stat. 9 (2012) 337–382], where weak forms of ergodicity are sufficient to ensure the validity of the Central Limit Theorem, we will need here strong ergodic properties: the existence of a spectral gap, hyperboundedness (or hypercontractivity). These properties are also discussed. Finally we give explicit examples.

This paper presents a practical and simple fully nonparametric multivariate smoothing procedure that adapts to the underlying smoothness of the true regression function. Our estimator is easily computed by successive application of existing base smoothers (without the need of selecting an optimal smoothing parameter), such as thin-plate spline or kernel smoothers. The resulting smoother has better out of sample predictive capabilities than the underlying base smoother, or competing structurally constrained models (MARS, GAM) for small dimension (3 ≤ d ≤ 7) and moderate sample size n ≤ 1000. Moreover our estimator is still useful when d > 10 and to our knowledge, no other adaptive fully nonparametric regression estimator is available without constrained assumption such as additivity for example. On a real example, the Boston Housing Data, our method reduces the out of sample prediction error by 20%. An R package ibr, available at CRAN, implements the proposed multivariate nonparametric method in R.

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

We use multivariate total positivity theory to exhibit new families of peacocks. As the authors of [F. Hirsch, C. Profeta, B. Roynette and M. Yor, Peacocks and associated martingales vol. 3. Bocconi-Springer (2011)], our guiding example is the result of Carr−Ewald−Xiao [P. Carr, C.-O. Ewald and Y. Xiao, Finance Res. Lett. 5 (2008) 162–171]. We shall introduce the notion of strong conditional monotonicity. This concept is strictly more restrictive than 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 are many random vectors which are strongly conditionally monotone (SCM). Indeed, we shall prove that multivariate totally positive of order 2 (MTP2) random vectors are SCM. As a consequence, stochastic processes with MTP2 finite-dimensional marginals are SCM. This family includes processes with independent and log-concave increments, and one-dimensional diffusions which have absolutely continuous transition kernels.

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

We consider dependence coefficients for stationary Markov chains. We emphasize on some equivalencies for reversible Markov chains. We improve some known results and provide a necessary condition for Markov chains based on Archimedean copulas to be exponential ρ-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 differential equations with intermittent upper barrier (RIBSDEs). These denote reflected BSDEs in which the upper barrier is only active on certain random time intervals. From the point of view of financial interpretation, RIBSDEs arise as pricing equations of game options with constrained callability. In a Markovian set-up we prove a convergence rate for a time-discretization scheme by simulation to an RIBSDE. We also characterize the solution of an RIBSDE as the largest viscosity subsolution of a related system of variational inequalities, and we establish the convergence of a deterministic numerical scheme for that problem. Due to the potentially very high dimension of the system of variational inequalities, this approach is not always practical. We thus subsequently prove a convergence 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 new finite difference operator ∇n, defined as a linear combination of left and right finite differences. We show that this operator ∇n plays a key role in a new Poincaré (spectral gap) inequality with respect to binomial weights, with the orthogonal Krawtchouk polynomials acting as eigenfunctions of the relevant operator. We briefly discuss the relationship of this operator to the problem of optimal transport of probability measures.

We study asymptotic behavior of Markov chain Monte Carlo (MCMC) procedures. Sometimes the performances of MCMC procedures are poor and there are great importance for the study of such behavior. In this paper we call degeneracy for a particular type of poor performances. We show some equivalent conditions for degeneracy. As an application, we consider 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 data augmentation (PX-DA) procedure is considered to be a remedy for it. In the sense of degeneracy, the PX-DA procedure is better than the DA procedure. However, when the number of categories is large, both procedures are degenerate and so the PX-DA procedure may not provide good estimate for the posterior distribution.

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

We consider the segmentation problem of Poisson and negative binomial (i.e. overdispersed Poisson) rate distributions. In segmentation, an important issue remains the choice of the number of segments. To this end, we propose a penalized -likelihood estimator where the penalty function 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. The performances of our criterion is assessed using simulated and real datasets in the RNA-seq data analysis context.

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

Consider a mean-reverting equation, generalized in the sense it is driven by a 1-dimensional centered Gaussian process with Hölder continuous paths on [0,T] (T> 0). Taking that equation in rough paths sense only gives local existence of the solution because the non-explosion condition is not satisfied in general. Under natural assumptions, by using specific methods, we show the global existence and uniqueness of the solution, its integrability, the continuity and differentiability of the associated Itô map, and we provide an Lp-converging approximation with a rate of convergence (p ≫ 1). The regularity of the Itô map ensures a large deviation 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 a generalized 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−Minkowski correspondence in ℝ2 states the existence of a homeomorphism between the probability 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 with perimeter 1. In this article, we bring out explicit formulas relating the border of a CCS to 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 probability measure operations, and reciprocally, the convolution of measures translates into a new notion of convolution of CCS. Additionally, we give a proof that a polygonal curve associated 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 the empirical process in statistics. Finally, we employ this correspondence to present models of smooth random CCS and simulations.

We study the estimation of the mean function of a continuous-time stochastic process and its derivatives. The covariance function of the process is assumed to be nonparametric and to satisfy mild smoothness conditions. Assuming that n independent realizations of the process are observed at a sampling design of size N generated by a positive density, we derive the asymptotic bias and variance of the local polynomial estimator as n,N increase to infinity. We deduce optimal sampling densities, optimal bandwidths, and propose a new plug-in bandwidth selection method. We establish the asymptotic performance of the plug-in bandwidth estimator and we compare, in a simulation study, its performance for finite sizes n,N to the cross-validation and the optimal bandwidths. A software implementation of the plug-in method is available in the R environment.