Optimal Variance--Gamma approximation on the second Wiener chaos

TL;DR

This paper establishes an optimal non-asymptotic bound for approximating a second Wiener chaos element by a Variance-Gamma distributed variable, extending the fourth moment theorem to this setting.

Contribution

It introduces a six moment theorem for Variance-Gamma approximation on the second Wiener chaos, expanding the scope of the Nourdin-Peccati methodology.

Findings

Derived a bound based on the maximum of the first six cumulants.

Extended the fourth moment theorem to Variance-Gamma approximation.

Applied results to the generalized Rosenblatt process at critical exponent.

Abstract

In this paper, we consider a target random variable $Y \sim \CVG$ distributed according to a centered Variance--Gamma distribution. For a generic random element $F=I_2(f)$ in the second Wiener chaos with $\E[F^2]= \E[Y^2]$ we establish a non-asymptotic optimal bound on the distance between $F$ and $Y$ in terms of the maximum of difference of the first six cumulants. This six moment theorem extends the celebrated optimal fourth moment theorem of I.\ Nourdin \& G.\ Peccati for normal approximation. The main body of our analysis constitutes a splitting technique for test functions in the Banach space of Lipschitz functions relying on the compactness of the Stein operator. The recent developments around Stein method for Variance--Gamma approximation by R.\ Gaunt play a significant role in our study. As an application we consider the generalized Rosenblatt process at the extreme critical exponent, first studied by S.\ Bai \& M.\ Taqqu.