Theory of Barnes Beta distributions

A new family of probability distributions beta M,N, M = 0 ... N, N is an element of IN 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.