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The Barnes $G$ function and its relations with sums and products of generalized Gamma convolution variables


 
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1. Title Title of document The Barnes $G$ function and its relations with sums and products of generalized Gamma convolution variables
 
2. Creator Author's name, affiliation, country Ashkan Nikeghbali; University of Zurich
 
2. Creator Author's name, affiliation, country Marc Yor; Universite Paris 6
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Barnes G-function, beta-gamma algebra, generalized gamma convolution variables, random matrices, characteristic polynomials of random unitary matrices
 
3. Subject Subject classification 60F99, 60E07, 60E10
 
4. Description Abstract We give a probabilistic interpretation for the Barnes $G$-function which appears in random matrix theory and in analytic number theory in the important moments conjecture due to Keating-Snaith for the Riemann zeta function, via the analogy with the characteristic polynomial of random unitary matrices. We show that the Mellin transform of the characteristic polynomial of random unitary matrices and the Barnes $G$-function are intimately related with products and sums of gamma, beta and log-gamma variables. In particular, we show that the law of the modulus of the characteristic polynomial of random unitary matrices can be expressed with the help of products of gamma or beta variables. This leads us to prove some non standard type of limit theorems for the logarithmic mean of the so called generalized gamma convolutions.
 
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7. Date (YYYY-MM-DD) 2009-09-23
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1488
 
10. Identifier Digital Object Identifier 10.1214/ECP.v14-1488
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 14
 
12. Language English=en
 
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