Related papers: A third-moment theorem and precise asymptotics for…
Consider $F$ an element of the second Wiener chaos with variance one. In full generality, we show that, for every integer $p\ge 1$, there exists $\eta_p>0$ such that if $\kappa_4(F)<\eta_p$ then the Malliavin derivative of $F$ admits a…
A general moment bound for sums of products of Gaussian vector's functions extending the moment bound in Taqqu (1977, Lemma 4.5) is established. A general central limit theorem for triangular arrays of nonlinear functionals of…
Using entropic inequalities from information theory, we provide new bounds on the total variation and 2-Wasserstein distances between a conditionally Gaussian law and a Gaussian law with invertible covariance matrix. We apply our results to…
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 $F_n$ towards a centered Gaussian random vector $N$, with given…
We prove a bound for the Wasserstein distance between vectors of smooth complex random variables and complex Gaussians in the framework of complex Markov diffusion generators. For the special case of chaotic eigenfunctions, this bound can…
The Rosenblatt process is a self-similar non-Gaussian process which lives in second Wiener chaos, and occurs as the limit of correlated random sequences in so-called \textquotedblleft non-central limit theorems\textquotedblright. It shares…
In this paper we obtain a rate of convergence in the central limit theorem for high order weighted Hermite variations of the fractional Brownian motion. The proof is based on the techniques of Malliavin calculus and the quantitative stable…
In this paper, we consider a target random variable $Y \sim \CVG$ distributed according to a centered Variance--Gamma distribution. For a generic random element $F=I_2(f)$ in the second Wiener chaos with $\E[F^2]= \E[Y^2]$ we establish a…
In Dolera, Gabetta and Regazzini [Ann. Appl. Probab. 19 (2009) 186-201] it is proved that the total variation distance between the solution $f(\cdot,t)$ of Kac's equation and the Gaussian density $(0,\sigma^2)$ has an upper bound which goes…
We study invariance principles and convergence to a Gaussian limit for stochastic series of the form $S(c,Z)=\sum_{m=1}^{\infty }\sum_{\alpha _{1}<...<\alpha _{m}}c(\alpha _{1},...,\alpha _{m})\prod_{i=1}^{m}Z_{\alpha _{i}}$ where $Z_{k}$,…
The aim of this paper is to establish the uniform convergence of the densities of a sequence of random variables, which are functionals of an underlying Gaussian process, to a normal density. Precise estimates for the uniform distance are…
Various quantum analogues of the central limit theorem, which is one of the cornerstones of probability theory, are known in the literature. One such analogue, due to Cushen and Hudson, is of particular relevance for quantum optics. It…
The Boltzmann equation for $d$-dimensional inelastic Maxwell models is considered to determine the collisional moments of second, third and fourth degree in a granular binary mixture. These collisional moments are exactly evaluated in terms…
We provide new bounds for the rate of convergence of the multivariate Central Limit Theorem in Wasserstein distances of order $p \geq 2$. In particular, we obtain what we conjecture to be the asymptotically optimal rate whenever the density…
We develop a new quantitative approach to a multidimensional version of the well-known {\it de Jong's central limit theorem} under optimal conditions, stating that a sequence of Hoeffding degenerate $U$-statistics whose fourth cumulants…
Consider a Gaussian stationary sequence with unit variance $X=\{X_k;k\in {\mathbb{N}}\cup\{0\}\}$. Assume that the central limit theorem holds for a weighted sum of the form $V_n=n^{-1/2}\sum^{n-1}_{k=0}f(X_k)$, where $f$ designates a…
This paper investigates a local central limit theorem for a normalized sequence of random variables belonging to a fixed order Wiener chaos and converging to the standard normal distribution. We prove, without imposing any additional…
In \cite{BNT}, a framework to prove almost sure central limit theorems for sequences $(G_n)$ belonging to the Wiener space was developed, with a particular emphasis of the case where $G_n$ takes the form of a multiple Wiener-It\^o integral…
Given a reference random variable, we study the solution of its Stein equation and obtain universal bounds on its first and second derivatives. We then extend the analysis of Nourdin and Peccati by bounding the Fortet-Mourier and…
We establish explicit bounds on the convex distance between the distribution of a vector of smooth functionals of a Gaussian field, and that of a normal vector with a positive definite covariance matrix. Our bounds are commensurate to the…