Theory of Barnes Beta distributions

A new family of probability distributions βM,N , M = 0 · · ·N, N ∈ N on the unit interval (0, 1] is defined by the Mellin transform. The Mellin transform of βM,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 βM,N is infinitely divisible. If M < N, − log βM,N is compound Poisson, if M = N, log βM,N is absolutely continuous. The integral moments of βM,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 β1,1 into a product of β −1 2,2s.