Brazilian Journal of Probability and Statistics

On free and classical type G distributions

Octavio Arizmendi, Ole E. Barndorff-Nielsen, and Víctor Pérez-Abreu

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There is a one-to-one correspondence between classical one-dimensional infinitely divisible distributions and free infinitely divisible distributions. In this work we study the free infinitely divisible distributions corresponding to the one-dimensional type G distributions. A new characterization of classical type G distributions is given first and the class of type A classical infinitely divisible distributions is introduced. The corresponding free type A distributions are studied and the role of a special symmetric beta distribution is shown as a building block for free type A distributions. It is proved that this symmetric beta distribution is the free multiplicative convolution of an arcsine distribution with the Marchenko–Pastur distribution.

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Braz. J. Probab. Stat. Volume 24, Number 2 (2010), 106-127.

First available in Project Euclid: 20 April 2010

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Variance mixtures of Gaussian free infinite divisibility free compound Poisson distribution transformation of Lévy measures free multiplicative convolution


Arizmendi, Octavio; Barndorff-Nielsen, Ole E.; Pérez-Abreu, Víctor. On free and classical type G distributions. Braz. J. Probab. Stat. 24 (2010), no. 2, 106--127. doi:10.1214/09-BJPS039.

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