| Título: | Theory of Barnes Beta distributions |
| Autores: | Ostrovsky, Dmitry; Independent researcher |
| Fecha: | 2013-01-03 |
| Publicador: | Electronic communications in probability |
| Fuente: |
 |
| Tipo: | Peer-reviewed Article |
| Tema: |
Multiple gamma function ; Infinite divisibility ; Selberg Integral ; Mellin transform 11M32; 30D05; 33B15; 41A60; 60E07; 60E10 |
| Descripción: | 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 $\beta^{-1}_{2, 2}s.$ |
| Idioma: | Inglés |
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