The Gamma Stein equation and non-central de Jong theorems

TL;DR

This paper develops new bounds for the Stein equation related to the Gamma distribution and applies them to improve quantitative approximation results for non-linear functionals of random fields across different probabilistic spaces.

Contribution

It introduces novel bounds for the Gamma Stein equation and extends quantitative de Jong theorems, Gamma approximation bounds on Poisson and Gaussian spaces, and a new Gamma approximation inequality.

Findings

Extended de Jong theorem for degenerate U-statistics.

Refined Gamma approximation bounds on Poisson space.

Strengthened Gamma bounds on Gaussian space.

Abstract

We study the Stein equation associated with the one-dimensional Gamma distribution, and provide novel bounds, allowing one to effectively deal with test functions supported by the whole real line. We apply our estimates to derive new quantitative results involving random variables that are non-linear functionals of random fields, namely: (i) a non-central quantitative de Jong theorem for sequences of degenerate U-statistics satisfying minimal uniform integrability conditions, significantly extending previous findings by de Jong (1990), Nourdin, Peccati and Reinert (2010) and D\"obler and Peccati (2016), (ii) a new Gamma approximation bound on the Poisson space, refining previous estimates by Peccati and Th\"ale (2013), and (iii) new Gamma bounds on a Gaussian space, strengthening estimates by Nourdin and Peccati (2009). As a by-product of our analysis, we also deduce a new inequality for Gamma approximations via exchangeable pairs, that is of independent interest.