Asymptotic Expansions in the CLT in Free Probability
G. P. Chistyakov, F. G\"otze
TL;DR
This paper develops Edgeworth type asymptotic expansions for sums of free random variables, providing detailed approximations of their distribution functions and densities under minimal moment conditions, with applications to free entropic distances.
Contribution
It introduces new Edgeworth expansions in free probability using analytic free convolution, extending classical CLT results to the free setting with minimal assumptions.
Findings
Derived Edgeworth expansions for free sums
Provided density expansion formulas
Applied results to free entropic distance calculations
Abstract
We prove Edgeworth type expansions for distribution functions of sums of free random variables under minimal moment conditions. The proofs are based on the analytic definition of free convolution. We apply these results to the expansion of densities to derive expansions for the free entropic distance of sums to the Wigner law.
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Taxonomy
TopicsRandom Matrices and Applications · Bayesian Methods and Mixture Models · Probability and Risk Models