Asymptotic Expansions in Free Limit Theorems

TL;DR

This paper extends classical asymptotic expansion techniques to free probability, deriving Edgeworth type expansions for sums of free random variables using their Cauchy transforms, under certain moment and support conditions.

Contribution

It generalizes the influence function approach to free probability, providing new Edgeworth expansions for sums of free variables with nine moments.

Findings

Derived Edgeworth expansions for free sums within (-2,2)

Established conditions on support and moments for expansions

Extended classical asymptotic methods to free probability context

Abstract

We study asymptotic expansions in free probability. In a class of classical limit theorems Edgeworth expansion can be obtained via a general approach using sequences of "influence" functions of individual random elements described by vectors of real parameters $(\varepsilon_1,..., \varepsilon_n)$, that is by a sequence of functions $h_n(\varepsilon_1,..., \varepsilon_n;t)$, $|\varepsilon_j| \leq \frac 1 {\sqrt n}$, $j=1,...,n$, $t\in {\mathbb R}$ (or ${\mathbb C}$) which are smooth, symmetric, compatible and have vanishing first derivatives at zero. In this work we expand this approach to free probability. As a sequence of functions $h_n(\varepsilon_1,..., \varepsilon_n;t)$ we consider a sequence of the Cauchy transforms of the sum $\sum_{j=1}^n \varepsilon_j X_j $, where $(X_j)_{j=1}^n$ are free identically distributed random variables with nine moments. We derive Edgeworth type expansions for distributions and densities (under the additional assumption that ${supp} (X_1) \subset [-\sqrt [3]n, \sqrt [3]n]$) of the sum $\frac 1 {\sqrt n} \sum_{j=1}^n X_j$ within the interval $(-2,2)$.