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 Γ i (introduced by Nourdin and Peccati in [13]), 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 [16], 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.

A bound on the Wasserstein-2 distance between linear combinations of independent random variables

“…• Combining Proposition 2.4 together with part (b) of Corollary 2.1, we obtain the fact that the convergence of ∆(F n ) to 0 is equivalent to the convergence of W 2 (F n , F ∞ ) to 0 when dim Q span{α 2 ∞,1 , · · · , α 2 ∞,q } = q. This complements the results contained in [25,2] (see in particular Theorem 2 of [2]). Moreover, recall that convergence of W 2 (F n , F ∞ ) to 0 is equivalent to convergence in distribution and convergence of the second moments.…”

“…In light of the coupling imposed by our Assumption it seems intuitively evident that W 2 (F n , F ∞ ) ought to be governed solely by the convergence rate of the approximating sequence of coefficients {α n,k } n,k≥1 towards {α ∞,k } 1≤k≤q . The main difficulty is to identify the correct norm for this convergence and, following on [2], we consider the quantity ∆(F n , F ∞ ) = k≥1 α 2 n,k q r=1 (α n,k − α ∞,r ) 2 .…”

Inequalities for integrals of modified Bessel functions and expressions involving them

“…Recently, Stein's method has been extended to variance-gamma (VG) approximation [12,14]. The VG distribution (also known as the generalized Laplace distribution [24]) is commonly used in financial mathematics [26], and has recently appeared in several papers in the probability literature as a limiting distribution [1,2,3]. This is in part due to the fact that the family of VG distributions is a rich one, with special or limiting cases that include, amongst others, the normal, gamma, Laplace, product of zero mean normals and difference of gammas [14,24].…”

On a new Sheffer class of polynomials related to normal product distribution

“…In the framework of the second Wiener chaos, it has been first proven in [23] using the method of complex analysis. For a rather general setup using the iterated Gamma operators and the Malliavin integration by parts formulae, see [7]. Also, for quantitative Berry-Essen estimates see the recent works [18,3], and [5,15] for the free counterpart statements.…”