Asymptotic Cram\'er type decomposition for Wiener and Wigner integrals

TL;DR

This paper explores asymptotic decompositions of Gaussian and semicircular distributions into sums of independent or free integrals, extending classical results to more general, asymptotic settings.

Contribution

It introduces asymptotic versions of Cramér's theorem for Wiener and Wigner integrals, broadening the understanding of distribution decompositions in probability theory.

Findings

Asymptotic decomposition results for Wiener integrals

Extension of Cramér's theorem to free probability

Validation of similar decomposition properties for semicircular distributions

Abstract

We investigate generalizations of the Cram\'er theorem. This theorem asserts that a Gaussian random variable can be decomposed into the sum of independent random variables if and only if they are Gaussian. We prove asymptotic counterparts of such decomposition results for multiple Wiener integrals and prove that similar results are true for the (asymptotic) decomposition of the semicircular distribution into free multiple Wigner integrals.