@article{ECP2445,
author = {Dmitry Ostrovsky},
title = {Theory of Barnes Beta distributions},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {18},
year = {2013},
keywords = {Multiple gamma function ; Infinite divisibility ; Selberg Integral ; Mellin transform},
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 $\beta^{-1}_{2, 2}s.$},
pages = {no. 59, 1-16},
issn = {1083-589X},
doi = {10.1214/ECP.v18-2445},
url = {http://ecp.ejpecp.org/article/view/2445}}