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The Central Limit Theorem for Free Additive Convolution

✍ Scribed by Vittorino Pata

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
852 KB
Volume
140
Category
Article
ISSN
0022-1236

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✦ Synopsis

Let [X n ] n=1 be a sequence of free, identically distributed random variables with common distribution +. Then there exist sequences [B n ] n=1 and [A n ] n=1 of positive and real numbers, respectively, such that sequence of random variables

converges in distribution to the semicircle law if and only if the function

is slowly varying in Karamata's sense. In other words, the free domain of attraction of the semicircle law coincides with the classical domain of attraction of the Gaussian. We prove an analogous result for normal domains of attraction in the sense of Linnik.

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