Convergence towards linear combinations of chi-squared random variables: a Malliavin-based approach
Abstract
We investigate the problem of finding necessary and sufficient conditions for convergence in distribution towards a general finite linear combination of independent chi-squared random variables, within the framework of random objects living on a fixed Gaussian space. Using a recent representation of cumulants in terms of the Malliavin calculus operators (introduced by Nourdin and Peccati in \cite{n-pe-3}), we provide conditions that apply to random variables living in a finite sum of Wiener chaoses. As an important by-product of our analysis, we shall derive a new proof and a new interpretation of a recent finding by Nourdin and Poly \cite{n-po-1}, concerning the limiting behaviour of random variables living in a Wiener chaos of order two. Our analysis contributes to a fertile line of research, that originates from questions raised by Marc Yor, in the framework of limit theorems for non-linear functionals of Brownian local times.
Cite
@article{arxiv.1409.5551,
title = {Convergence towards linear combinations of chi-squared random variables: a Malliavin-based approach},
author = {Ehsan Azmoodeh and Giovanni Peccati and Guillaume Poly},
journal= {arXiv preprint arXiv:1409.5551},
year = {2014}
}