Local limit theorems in free probability theory

Abstract

In this paper, we study the superconvergence phenomenon in the free central limit theorem for identically distributed, unbounded summands. We prove not only the uniform convergence of the densities to the semicircular density but also their $L^p$-convergence to the same limit for $p>1/2$. Moreover, an entropic central limit theorem is obtained as a consequence of the above results.

Published in at http://dx.doi.org/10.1214/09-AOP505 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• On the Law of Addition of Random Matrices
• On superconvergence of sums of free random variables
• Semicircularity, Gaussianity and Monotonicity of Entropy

Cited by in corpus (12)

• Superconvergence to freely infinitely divisible distributions
• An inequality for the distance between densities of free convolutions
• On the support of the free additive convolution
• Rate of Convergence in the entropic free CLT
• New limit theorems related to free multiplicative convolution
• Asymptotic Expansions in Free Limit Theorems
• The Brown measure of the sum of a self-adjoint element and an elliptic element
• Berry-Esseen bounds for the multivariate $\mathcal{B}$-free CLT and operator-valued matrices
• Asymptotic Expansions in the CLT in Free Probability
• Uniform Convergence and the Central Limit Theorem
• Superconvergence and regularity of densities in free probability
• Edgeworth-type expansion in the entropic free CLT