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 $\Gamma _{i}$ (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.

MSC 2010: 60F05; 60G15; 60H07

Appendix

Lemma 6

Let $\{a_{k}\}_{k\in \mathbb{N}}$ and $\{b_{k}\}_{k\in \mathbb{N}}$ be two sequences in $l^{1}(\mathbb{N})$ such that for all p ≥ 1 we have

$$\displaystyle{ \sum _{k\in \mathbb{N}}a_{k}^{p} =\sum _{ k\in \mathbb{N}}b_{k}^{p}. }$$

(43)

Then, there exists a permutation π on natural numbers $\mathbb{N}$ such that a _k = b _π(k) for all k ≥ 1.

Proof

Let $\mathbb{R}[X]$ denote the ring of all polynomials over real line. Then, relation (43) implies that for any polynomial $P \in \mathbb{R}[X]$, we have

$$\displaystyle{ \sum _{k\in \mathbb{N}}a_{k}P(a_{k}) =\sum _{k\in \mathbb{N}}b_{k}P(b_{k}). }$$

(44)

Let $M:=\max \{\Vert a\Vert _{l^{1}(\mathbb{N})},\Vert b\Vert _{l^{1}(\mathbb{N})}\} <\infty$. Then by a density argument, for any continuous function $\varphi \in C([-M,M])$, we obtain

$$\displaystyle{ \sum _{k\in \mathbb{N}}a_{k}\varphi (a_{k}) =\sum _{k\in \mathbb{N}}b_{k}\varphi (b_{k}). }$$

(45)

For any $i \in \mathbb{N}$, we can now choose a continuous function $\varphi$ such that $\varphi (a_{i}) = 1$ and $\varphi = 0$ on the set $\{a_{j}\vert a_{j}\neq a_{i}\} \cup \{ b_{j}\vert b_{j}\neq a_{i}\}$. This implies that, for some integer k _i , we have $a_{i} = b_{k_{i}}$. It is now sufficient to take π(i) = k _i .