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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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 20 April 2010

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1271770266

Digital Object Identifier
doi:10.1214/09-BJPS039

Mathematical Reviews number (MathSciNet)
MR2643561

Zentralblatt MATH identifier
1209.62010

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

Citation

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. https://projecteuclid.org/euclid.bjps/1271770266.


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