Asymptotic Expansions in Free Limit Theorems

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 | ≤ 1 n , j = 1, . . . , n, t ∈ A ⊂ R (or 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 ∑n j=1 εjXj , where (Xj) n j=1 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 1 √ n ∑n j=1 Xj within the interval (−2, 2).