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Theory of Barnes Beta Distributions
A new family of probability distributions
on the unit interval is defined by the Mellin
transform. The Mellin transform of 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 is infinitely divisible. If
is compound Poisson, if is
absolutely continuous. The integral moments of 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 into a product of
Comment: 15 pages, published version (removed Th. 4.5 and Section 5, updated
references
Binomial Identities and Moments of Random Variables
We give unified simple proofs of some binomial identities, by using an elementary identity on moments of random variables
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