Electronic Communications in Probability

Theory of Barnes Beta distributions

Dmitry Ostrovsky

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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 $\beta^{-1}_{2, 2}s.$

Article information

Electron. Commun. Probab., Volume 18 (2013), paper no. 59, 16 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M32: Multiple Dirichlet series and zeta functions and multizeta values
Secondary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX] 33B15: Gamma, beta and polygamma functions 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.) [See also 30E15] 60E07: Infinitely divisible distributions; stable distributions 60E10: Characteristic functions; other transforms

Multiple gamma function Infinite divisibility Selberg Integral Mellin transform

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Ostrovsky, Dmitry. Theory of Barnes Beta distributions. Electron. Commun. Probab. 18 (2013), paper no. 59, 16 pp. doi:10.1214/ECP.v18-2445. https://projecteuclid.org/euclid.ecp/1465315598

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