Asymptotic Expansions in Free Limit Theorems

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 $(-2,2)$.