We define a covariance-type operator on Wiener space: for F and G two random variables in the Gross–Sobolev space D1,2 of random variables with a square-integrable Malliavin derivative, we let ΓF,G≔〈DF,−DL−1G〉, where D is the Malliavin derivative operator and L−1 is the...

We consider sequences of random variables of the type , n=1, where is a d-dimensional Gaussian process and is a measurable function. It is known that, under certain conditions on f and the covariance function r of X, Sn converges in distribution to a normal variable S. In the present paper we...

Let X1,X2,… be a sequence of i.i.d. random variables, with mean zero and variance one and let Sn=(X1+⋯+Xn)/n. An old and celebrated result of Prohorov (1952) asserts that Sn converges in total variation to the standard Gaussian distribution if and only if Sn0 has an absolutely continuous...

Let {Fn} be a sequence of random variables belonging to a finite sum of Wiener chaoses. Assume further that it converges in distribution towards F∞ satisfying V ar(F∞)0. Our first result is a sequential version of a theorem by Shigekawa (1980) [23]. More precisely, we prove, without...

In this paper, we study almost sure central limit theorems for sequences of functionals of general Gaussian fields. We apply our result to non-linear functions of stationary Gaussian sequences. We obtain almost sure central limit theorems for these non-linear functions when they converge in law...

By combining the findings of two recent, seminal papers by Nualart, Peccati and Tudor, we get that the convergence in law of any sequence of vector-valued multiple integrals Fn towards a centered Gaussian random vector N, with given covariance matrix C, is reduced to just the convergence of: (i)...

The problem of absolute continuity for a class of SDEs driven by a real fractional Brownian motion of any Hurst index is addressed. First, we give an elementary proof of the fact that when the diffusion coefficient does not vanish, the solution to the SDE has a positive density for all t0....

Continuing the analysis initiated by Lachièze-Rey and Peccati (2013), we use contraction operators to study the normal approximation of random variables having the form of a U-statistic written on the points in the support of a random Poisson measure. Applications are provided to subgraph...

We study Hoeffding decomposable exchangeable sequences with values in a finite set D={d1,…,dK}. We provide a new combinatorial characterization of Hoeffding decomposability and use this result to show that, for every K≥3, there exists a class of neither Pólya nor i.i.d. D-valued...

We use the concept of time-space chaos (see Peccati (Ann. Inst. Poincaré 37(5) (2001) 607; Prépublication n. 648 du Laboratoire de Probabilités et Modèles Aléatoires de l'Université Paris VI; Chaos Brownien d'espace-temps, décompositions de Hoeffding et problèmes de convergence associés,...