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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
Preservation of the Borel class under open- functions
Let be a Borel subset of the Cantor set \textbf{C} of additive or
multiplicative class and be a continuous function with
compact preimages of points onto If the image of
every clopen set is the intersection of an open and a closed set, then
is a Borel set of the same class. This result generalizes similar results for
open and closed functions.Comment: 5 page
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