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New limit theorems related to free multiplicative convolution

Noriyoshi SakumaHiroaki Yoshida — 2013

Studia Mathematica

Let ⊞, ⊠, and ⊎ be the free additive, free multiplicative, and boolean additive convolutions, respectively. For a probability measure μ on [0,∞) with finite second moment, we find a scaling limit of ( μ N ) N as N goes to infinity. The -transform of its limit distribution can be represented by Lambert’s W-function. From this, we deduce that the limiting distribution is freely infinitely divisible, like the lognormal distribution in the classical case. We also show a similar limit theorem by replacing free...

Generalized q-deformed Gaussian random variables

Marek BożejkoHiroaki Yoshida — 2006

Banach Center Publications

We produce generalized q-Gaussian random variables which have two parameters of deformation. One of them is, of course, q as for the usual q-deformation. We also investigate the corresponding Wick formulas, which will be described by some joint statistics on pair partitions.

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