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2013 Theory of Barnes Beta distributions
Dmitry Ostrovsky
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Electron. Commun. Probab. 18: 1-16 (2013). DOI: 10.1214/ECP.v18-2445


A new family of probability distributions $\beta_{M, N},$ $M=0\cdots N,$ $N\in\mathbb{N}$ on the unit interval $(0, 1]$ is defined by the Mellin transform. The Mellin transform of $\beta_{M,N}$ is characterized in terms of products of ratios of Barnes multiple gamma functions, shown to satisfy a functional equation, and a Shintani-type infinite product factorization. The distribution $\log\beta_{M, N}$ is infinitely divisible. If $M<N,$ $-\log\beta_{M, N}$ is compound Poisson, if $M=N,$ $\log\beta_{M, N}$ is absolutely continuous. The integral moments of $\beta_{M, N}$ are expressed as Selberg-type products of multiple gamma functions. The asymptotic behavior of the Mellin transform is derived and used to prove an inequality involving multiple gamma functions and establish positivity of a class of alternating power series. For application, the Selberg integral is interpreted probabilistically as a transformation of $\beta_{1, 1}$ into a product of $\beta^{-1}_{2, 2}s.$


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Dmitry Ostrovsky. "Theory of Barnes Beta distributions." Electron. Commun. Probab. 18 1 - 16, 2013.


Accepted: 12 July 2013; Published: 2013
First available in Project Euclid: 7 June 2016

zbMATH: 1300.60032
MathSciNet: MR3084570
Digital Object Identifier: 10.1214/ECP.v18-2445

Primary: 11M32
Secondary: 30D05, 33B15, 41A60, 60E07, 60E10


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