Berry–Esseen for Free Random Variables

Abstract

An analogue of the Berry-Esseen inequality is proved for the speed of convergence of free additive convolutions of bounded probability measures. The obtained rate of convergence is of the order n −1/2 , the same as in the classical case. An example with binomial measures shows that this estimate cannot be improved without imposing further restrictions on convolved measures.

Asymptotic expansions in the CLT in free probability

“…The upper bound (2.19) sharpens previous results obtained by the authors [22] and V. Kargin [26]. Theorem 2.1 and Corollary 2.2 are free analogues of Esseen's inequality in classical probability theory (see [36], p. 112-120).…”

“…A free analogue of this classical result was proved by Voiculescu [39] for bounded free random variables and later generalized by Maassen [30] to unbounded random variables. Other generalizations can be found in [10], [11], [22], [26]- [28], [35], [45], [46]. When the assumption of independence is replaced by the freeness of the noncommutative random variables X 1 , X 2 , .…”

“…An analog of the Berry-Esseen inequality was proved for the semicircle approximation in [22]. For related results see [26].…”

Uniform Convergence and the Free Central Limit Theorem

“…A number of free analogues of Berry-Esseen have been proven with respect to this function. Most notably, in [5], it was shown that for µ a measure with bounded support,…”

Operator-Valued and Multivariate Free Berry-Esseen Theorems

“…where C is a constant and ρ is the absolute third moment of the variables X i . The question for a free analogue of the Berry-Esseen estimate in the case of one random variable was answered by Christyakov and Götze in [2] (and independently, under the more restrictive assumption of compact support of the X i , by Kargin [10]): If (X i ) i∈N is a sequence of free and identically distributed variables with mean 0 and variance 1, then the distance between S n := 1 √ n (X 1 + · · · + X n ) and a semicircular variable s of mean 0 and variance 1 can be estimated as…”