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Theory of Barnes Beta distributions


 
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1. Title Title of document Theory of Barnes Beta distributions
 
2. Creator Author's name, affiliation, country Dmitry Ostrovsky; Independent researcher; United States
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Multiple gamma function ; Infinite divisibility ; Selberg Integral ; Mellin transform
 
3. Subject Subject classification 11M32; 30D05; 33B15; 41A60; 60E07; 60E10
 
4. Description Abstract 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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7. Date (YYYY-MM-DD) 2013-07-12
 
8. Type Status & genre Peer-reviewed Article
 
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9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/2445
 
10. Identifier Digital Object Identifier 10.1214/ECP.v18-2445
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 18
 
12. Language English=en en
 
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