Local limit theorems in free probability theory

Details Local limit theorems in free probability theory Local limit theorems in free probability theory

2010-10-18

arxiv.org/abs/1010.3579v1

Annals of Probability 2010, Vol. 38, No. 4, 1492-1506

Mathematics

Probability

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

Scientific paper

10.1214/09-AOP505

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.

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