In this paper we prove a Central Limit Theorem forstandard kernel estimates of the invariant density of one-dimensionaldynamical systems. The two main steps of the proof of this theorem are the following: the study of rate of convergencefor the variance of the estimator and a variation on the Lindeberg–Riomethod. We also give an extension in the case of weaklydependent sequences in a sense introduced by Doukhan and Louhichi.

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.

Consider a random environment in ${\mathbb Z}^d$ given by i.i.d. conductances.In this work, we obtain tail estimates for the fluctuations about themean for the following characteristics of the environment: the effective conductance between opposite faces of a cube, the diffusion matrices of periodized environmentsand the spectral gap of the random walk in a finite cube.

As in preceding papers inwhich we studied the limits of penalized 1-dimensional Wienermeasures with certain functionals Γt, we obtain here theexistence of the limit, as t → ∞, of d-dimensional Wienermeasures penalized by a function of the maximum up to time t ofthe Brownian winding process (for d = 2), or in {d}≥ 2dimensions for Brownian motionprevented to exit a cone before time t. Various extensions of these multidimensional penalisations arestudied, and the limit laws are described. Throughout this paper, the skew-product decomposition ofd-dimensional Brownian motion plays an important role.

Let F be a distribution function (d.f) in the domain of attraction of an extreme value distribution ${ H_{\gamma} }$; it is well-known that Fu(x), where Fu is the d.f of the excesses over u, converges, when u tends to s+(F), the end-point of F, to $G_{\gamma}(\frac{x}{\sigma(u)})$, where $G_{\gamma}$ is the d.f. of the Generalized Pareto Distribution. We provide conditions that ensure that there exists, for $\gamma >-1$, a function Λ which verifies $\lim_{u \rightarrow s_+(F)} \Lambda (u) =\gamma$ and is such that $\Delta(u)= \sup_{x \in[0,s_+(F)-u[} |\bar{F}_u(x) - \bar{G}_{\Lambda(u)} (x/ \sigma(u))| $converges to 0 faster than $d(u)=\sup_{x \in[0,s_+(F)-u[} |\bar{F}_u(x) - \bar{G}_{\gamma}(x/ \sigma(u))|$.

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.

In this paper a multi-scaled diffusion-approximation theorem is proved so as to unify various applications in wave propagation in random media: transmission of optical modes through randomplanar waveguides; time delay in scattering for the linear wave equation; decay of the transmission coefficient for large lengths with fixed output and phase difference in weakly nonlinear randommedia.

The P.O.T. (Peaks-Over-Threshold) approachconsists of using the Generalized ParetoDistribution (GPD)to approximate the distribution of excesses over a threshold.We use the probability-weighted momentsto estimate the parameters of the approximating distribution.We study the asymptotic behaviour ofthese estimators (in particular their asymptotic bias) and also thefunctional bias of the GPD as an estimate of thedistribution function of the excesses. We adapt penultimateapproximation results to the case where parameters are estimated.

We study the estimation of a linear integral functional of a distribution F, using i.i.d. observations which density is a mixture of a family of densities k(.,y) under F. We examine the asymptotic distribution of the estimator obtained by plugging the non parametric maximum likelihood estimator (NPMLE) of F in the functional. A problem here is that usually, the NPMLE does not dominateF.
Our main aim here is to show that this can be overcome by considering a convex combination of F and the NPMLE.

In this article, our aim is to estimate the successive derivatives of the stationarydensity f of a strictly stationary and β-mixing process (Xt)t≥0. This process is observed at discrete timest = 0,Δ,...,nΔ. Thesampling interval Δ can be fixed or small. We use a penalizedleast-square approach to compute adaptive estimators. If the derivativef(j) belongs to the Besov space \hbox{$\rond{B}_{2,\infty}^{\alpha}$}$B_{\mathrm{2}\mathit{,}\mathrm{\infty }}^{\mathit{\alpha }}$, then our estimator converges at rate (nΔ)−α/(2α+2j+1). Then we consider a diffusion with known diffusioncoefficient. We use the particular form of the stationary density to compute an adaptiveestimator of its first derivative f′. When the sampling intervalΔ tends to 0, and when the diffusion coefficient is known, theconvergence rate of our estimator is (nΔ)−α/(2α+1). When the diffusion coefficient is known, we alsoconstruct a quotient estimator of the drift for low-frequency data.

In this paper we study asymptotic behavior of convexrearrangements of Lévy processes. In particular weobtain Glivenko-Cantelli-type strong limit theorems for the convexifications when the corresponding Lévy measureis regularly varying at + with exponent α ∈ (1,2).

We consider the nonparametric regression estimation problem of recovering an unknownresponse function f on the basis of spatially inhomogeneous data when thedesign points follow a known density g with a finite number ofwell-separated zeros. In particular, we consider two different cases: wheng has zeros of a polynomial order and when g has zerosof an exponential order. These two cases correspond to moderate and severe data losses,respectively. We obtain asymptotic (as the sample size increases) minimax lower bounds forthe L2-risk when f is assumed to belong to aBesov ball, and construct adaptive wavelet thresholding estimators of fthat are asymptotically optimal (in the minimax sense) or near-optimal within alogarithmic factor (in the case of a zero of a polynomial order), over a wide range ofBesov balls. The spatially inhomogeneous ill-posed problem that we investigate isinherently more difficult than spatially homogeneous ill-posed problems like,e.g., deconvolution. In particular, due to spatial irregularity,assessment of asymptotic minimax global convergence rates is a much harder task than thederivation of asymptotic minimax local convergence rates studied recently in theliterature. Furthermore, the resulting estimators exhibit very different behavior andasymptotic minimax global convergence rates in comparison with the solution of spatiallyhomogeneous ill-posed problems. For example, unlike in the deconvolution problem, theasymptotic minimax global convergence rates are greatly influenced not only by the extentof data loss but also by the degree of spatial homogeneity of f.Specifically, even if 1/g is non-integrable, one can recoverf as well as in the case of an equispaced design (in terms ofasymptotic minimax global convergence rates) when it is homogeneous enough since theestimator is “borrowing strength” in the areas where f is adequatelysampled.

We deal with the problem of choosing a piecewise constant estimator of a regression function s mapping $\mathcal{X}$ into $\mathbb{R}$ .We consider a non Gaussian regression framework with deterministic design points, and we adopt the non asymptotic approach of model selection via penalization developed by Birgé and Massart. Given a collection of partitions of $\mathcal{X}$ , with possibly exponential complexity,and the corresponding collection of piecewise constant estimators, we propose a penalized least squares criterion which selects a partition whose associated estimator performs approximately as well as the best one, in the sense that its quadratic risk is close to the infimum of the risks. The risk bound we provide is non asymptotic.

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.

In this paper we are interested in the estimation of a density − defined on a compact interval ofℝ− from n independent andidentically distributed observations. In order to avoid boundary effect, beta kernelestimators are used and we propose a procedure (inspired by Lepski’s method) in order toselect the bandwidth. Our procedure is proved to be adaptive in an asymptotically minimaxframework. Our estimator is compared with both the cross-validation algorithm and theoracle estimator using simulated data.

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.

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.

We generalize a theorem of Shao [Proc. Amer. Math. Soc.123 (1995) 575–582] on the almost-sure limiting behavior of the maximum of standardized random walk increments to multidimensional arrays of i.i.d. random variables. The main difficulty is the absence of an appropriate strong approximation result in the multidimensional setting. The multiscale statistic under consideration was used recently for the selection of the regularization parameter in a number of statistical algorithms as well as for the multiscale signal detection.

Unbiased risk estimation, à la Stein, is studied for infinitely divisible laws with finite second moment.

We prove a strong law of large numbers for moving averages of independent, identically distributed random variables with certain subexponential distributions. These random variables show a behavior that can be considered intermediate between the classical strong law and the Erdös-Rényi law. We further show that the difference from the classical behavior is due to the influence of extreme terms.