Theory of Barnes Beta Distributions

A new family of probability distributions $β_{M, N},$ $M=0\cdots N,$ $N\in\mathbb{N}$ on the unit interval $(0, 1]$ is defined by the Mellin transform. The Mellin transform of $β_{M, N}$ 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 $\logβ_{M, N}$ is infinitely divisible. If $M<N,$ $-\logβ_{M, N}$ is compound Poisson, if $M=N,$ $\logβ_{M, N}$ is absolutely continuous. The integral moments of $β_{M, N}$ 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 $β_{1, 1}$ into a product of $β^{-1}_{2, 2}s.$

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