We consider sequences of random variables of the type $S_n= n^{-1/2} \sum_{k=1}^n f(X_k)$, $n\geq 1$, where $X=(X_k)_{k\in \Z}$ is a $d$-dimensional Gaussian process and $f: \R^d \rightarrow \R$ is a measurable function. It is known that, under certain conditions on $f$ and the covariance...

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...

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...

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...

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...

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)...

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...

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....