2014/08/06 by F. Götze, Götze, F., A. Reshetenko +1
Mathematics · #46L54 #60B10 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:46L54 #msc:60B10
paper · pdf · doi:10.48550/arxiv.1408.1360
We remove the condition of bounded support of measures and require nine moments. Some typos have been corrected
arxiv created 2015/02/03 · arxiv updated 2015/02/05
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 (ε1,..., εn), that is by a sequence of functions hn(ε1,..., εn;t), |εj| ≤ \frac 1 √ n, j=1,...,n, t∈ \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 hn(ε1,..., εn;t) we consider a sequence of the Cauchy transforms of the sum ∑j=1n εj Xj , where (Xj)j=1n are free identically distributed random variables with nine moments. We derive Edgeworth type expansions for distributions and densities (under the additional assumption that supp (X1) ⊂ [-√ [3]n, √ [3]n]) of the sum \frac 1 √ n ∑j=1n Xj within the interval (-2,2).