Electronic Communications in Probability
- Electron. Commun. Probab.
- Volume 14 (2009), paper no. 39, 396-411.
The Barnes $G$ function and its relations with sums and products of generalized Gamma convolution variables
Ashkan Nikeghbali and Marc Yor
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.
Article information
Source
Electron. Commun. Probab., Volume 14 (2009), paper no. 39, 396-411.
Dates
Accepted: 23 September 2009
First available in Project Euclid: 6 June 2016
Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465234748
Digital Object Identifier
doi:10.1214/ECP.v14-1488
Mathematical Reviews number (MathSciNet)
MR2545290
Zentralblatt MATH identifier
1189.60073
Subjects
Primary: 60F99: None of the above, but in this section
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60E10: Characteristic functions; other transforms
Keywords
Barnes G-function beta-gamma algebra generalized gamma convolution variables random matrices characteristic polynomials of random unitary matrices
Rights
This work is licensed under aCreative Commons Attribution 3.0 License.
Citation
Nikeghbali, Ashkan; Yor, Marc. The Barnes $G$ function and its relations with sums and products of generalized Gamma convolution variables. Electron. Commun. Probab. 14 (2009), paper no. 39, 396--411. doi:10.1214/ECP.v14-1488. https://projecteuclid.org/euclid.ecp/1465234748