Stable Laws and Domains of Attraction in Free Probability Theory

Abstract

In this paper we determine the distributional behavior of sums of free (in the sense of Voiculescu) identically distributed, infinitesimal random variables. The theory is shown to parallel the classical theory of independent random variables, though the limit laws are usually quite different. Our work subsumes all previously known instances of weak convergence of sums of free, identically distributed random variables. In particular, we determine the domains of attraction of stable distributions in the free theory. These freely stable distributions are studied in detail in the appendix, where their unimodality and duality properties are demonstrated.

Infinitely divisible distributions for rectangular free convolution: classification and matricial interpretation

“…To prove the last part of the corollary, recall the fact from [GK54], which is also recalled in [BPB99], that for all sequence (η n ) of probability measures on the real line, for all sequence (k n ) of integers tending to +∞, for all real number a and all positive measure finite H on the real line, we have the equivalence…”

“…As recalled in Theorem 3.3 of [BPB99], it is proved in [GK54] that when (ν n ) is a sequence of symmetric probability measures on the real line and (k n ) is a sequence of integers tending to infinity, the sequence ν * kn n converges weakly to a * -infinitely divisible distribution if and only if the sequence knt 2 1+t 2 dν n (t) of positive finite measures converges weakly to its Lévy measure. By the theorem 2.5, we know that it will be the case if the sequence ν ⊞ λ kn n converges weakly to the image of the * -infinitely divisible distribution by the rectangular Bercovici-Pata bijection.…”

Weingarten integration over noncommutative homogeneous spaces

“…Our purpose now will be to establish some liberation results, in the sense of the Bercovici-Pata bijection [7].…”

Unimodality of Boolean and Monotone Stable Distributions

“…The aspect of domains of attraction for these distributions are studied in [W12,AW13] and [BP99], respectively.…”

“…classical stable random variables X, Y , and at the same time, it is the law of the "noncommutative quotient" X´1 {2 Y X´1 {2 of free random variables X, Y having the same free stable law [BP99,AH14].…”