Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos

TL;DR

This paper develops an asymptotic expansion theory for vector-valued random variables using Stein-Malliavin techniques, providing second-order density approximations and illustrating with examples.

Contribution

It introduces a novel approach combining Fourier, Stein, and Malliavin methods to derive second-order density expansions for vector-valued sequences.

Findings

Derived second-order asymptotic density expansion

Unified Fourier and Stein-Malliavin approach

Validated results with multiple examples

Abstract

We develop the asymptotic expansion theory for vector-valued sequences (F N) N $\ge$1 of random variables in terms of the convergence of the Stein-Malliavin matrix associated to the sequence F N. Our approach combines the classical Fourier approach and the recent theory on Stein method and Malliavin calculus. We find the second order term of the asymptotic expansion of the density of F N and we illustrate our results by several examples. 2010 AMS Classification Numbers: 62M09, 60F05, 62H12