Theory of Barnes Beta Distributions

TL;DR

This paper introduces a new family of probability distributions called Barnes Beta distributions, characterized by their Mellin transforms involving Barnes gamma functions, with properties like infinite divisibility and applications to the Selberg integral.

Contribution

The paper defines Barnes Beta distributions via Mellin transforms, explores their properties, and connects them to special functions and integrals, providing new tools for probabilistic and analytical applications.

Findings

Barnes Beta distributions are infinitely divisible.

Logarithm of Barnes Beta distributions exhibits specific divisibility properties.

The distributions relate to the Selberg integral through probabilistic transformations.

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 $\beta^{-1}_{2, 2}s.$