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 $({\epsilon }_1,...,{\epsilon }_n)$, that is by a sequence of functions $h_n({\epsilon }_1,...,{\epsilon }_n;t)$, $|{\epsilon }_j|\le \frac{1}{\sqrt{n}}$, $j=1,...,n$, $t\in \mathbb{R}$ (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 $h_n({\epsilon }_1,...,{\epsilon }_n;t)$ we consider a sequence of the Cauchy transforms of the sum ${\sum }_{j=1}^n{\epsilon }_j{X}_j$, 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 $(-2,2)$.

We remove the condition of bounded support of measures and require nine moments. Some typos have been corrected

References in corpus (2)

• Local limit theorems in free probability theory
• Analyzing Answers in Threaded Discussions using a Role-Based Information Network