Non-central limit of densities of some functionals of Gaussian processes

TL;DR

This paper proves that the densities of certain nonlinear Gaussian functionals converge to a Gamma distribution density, introducing a new density formula and applying Malliavin calculus and Stein's method for precise density approximation.

Contribution

It introduces a novel density formula for Markov diffusion generators and applies it to establish Gamma density convergence for Gaussian functionals, including in infinite chaos cases.

Findings

Densities of nonlinear Gaussian functionals converge to Gamma densities.

New density formula for Markov diffusion generators facilitates Gamma approximation.

Effective bounds on density differences using moments up to order four.

Abstract

We establish the convergence of the densities of a sequence of nonlinear functionals of an underlying Gaussian process to the density of a Gamma distribution. The key idea of our work is a new density formula for random variables in the setting of Markov diffusion generators, which yields a special representation for the density of a Gamma distribution. Via this representation, we are able to provide precise estimates on the distance between densities while developing the techniques of Malliavin calculus and Stein's method suitable to Gamma approximation at the density level. We first focus our study on the case of random variables living in a fixed Wiener chaos of an even order for which the bound for the difference of the densities can be dominated by a linear combination of moments up to order four. We then study the case of general Gaussian functionals with possibly infinite chaos expansion. Finally, we provide an application to random variables living in the second Wiener chaos.