Bilateral Gamma Approximation in Weiner Space

TL;DR

This paper introduces a new method for approximating functionals of Gaussian processes using bilateral-gamma distributions, providing sharper error bounds and convergence criteria in the Wiener chaos framework.

Contribution

It develops a novel Malliavin-Stein approach based on integral operators for bilateral-gamma approximation, improving error bounds and establishing convergence conditions in the second Wiener chaos.

Findings

Error bounds for bilateral-gamma approximation are derived.

Convergence in distribution to bilateral-gamma is characterized by cumulant convergence.

Applications include approximation of homogeneous sums and U-statistics.

Abstract

This paper deals with bilateral-gamma (BG) approximation to functionals of an isonormal Gaussian process. We use Malliavin-Stein method to obtain the error bounds for the smooth Wasserstein distance. As by-products, the error bounds for variance-gamma (V G), Laplace, gamma and normal approximations are presented. Our approach is new in the sense that the Stein equation is based on integral operators rather than diferential operators commonly used in the literature. Some of our bounds are sharper than the existing ones. For the approximation of a random element from the second Wiener chaos to a BG distribution, the bounds are obtained in terms of their cumulants. Using this result, we show that a sequence of random variables (rvs) in the second Wiener chaos converges in distribution to a BG rv if their cumulants of order two to six converge. As an application of our results, we consider an approximation of homogeneous sums of independent rvs to a BG distribution, and mention some related limit theorems also. Finally, an approximation of a U-statistic to the BG distribution is discussed.