Convergence Towards Linear Combinations of Chi-Squared Random Variables: A Malliavin-Based Approach

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 \(\Gamma _{i}\) (introduced by Nourdin and Peccati, J. Appl. Funct. Anal. 258(11), 3775–3791, 2010), 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 (Electron. Commun. Probab. 17(36), 1–12, 2012), concerning the limiting behavior 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.