Home | Contents | Submissions, editors, etc. | Login | Search | EJP
 Electronic Communications in Probability > Vol. 14 (2009) > Paper 39 open journal systems 


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

Ashkan Nikeghbali, University of Zurich
Marc Yor, Universite Paris 6


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.


Full text: PDF

Pages: 396-411

Published on: September 23, 2009
Bibliography
  1. Adamchik, V. S. On the Barnes function. Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation, 15--20 (electronic), ACM, New York, 2001. MR2044587 (2005a:33001)
  2. Andrews, George E.; Askey, Richard; Roy, Ranjan. Special functions.Encyclopedia of Mathematics and its Applications, 71. Cambridge University Press, Cambridge, 1999. xvi+664 pp. ISBN: 0-521-62321-9; 0-521-78988-5 MR1688958 (2000g:33001)
  3. Barnes,E.W: The theory of the G-function, Quart. J. Pure Appl. Math. 31 (1899), 264--314.
  4. Biane, Philippe; Pitman, Jim; Yor, Marc. Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions. Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 4, 435--465 (electronic). MR1848256 (2003b:11083)
  5. Biane, Ph.; Yor, M. Valeurs principales associées aux temps locaux browniens.(French) [Principal values associated with Brownian local times] Bull. Sci. Math. (2) 111 (1987), no. 1, 23--101. MR0886959 (88g:60188)
  6. Bondesson, Lennart. Generalized gamma convolutions and related classes of distributions and densities.Lecture Notes in Statistics, 76. Springer-Verlag, New York, 1992. viii+173 pp. ISBN: 0-387-97866-6 MR1224674 (94g:60031)
  7. Bottcher, A. ; Silbermann. B: Analysis of Toeplitz operators, Springer-Verlag, Berlin. Second Edition (2006).
  8. Bourgade, P.; Hughes, C. P.; Nikeghbali, A.; Yor, M. The characteristic polynomial of a random unitary matrix: a probabilistic approach. Duke Math. J. 145 (2008), no. 1, 45--69. MR2451289 (2009j:60011)
  9. Carmona, Philippe; Petit, Frédérique; Yor, Marc. On the distribution and asymptotic results for exponential functionals of Lévy processes. Exponential functionals and principal values related to Brownian motion, 73--130, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 1997. MR1648657 (99h:60144)
  10. Chaumont, L.; Yor, M. Exercises in probability.A guided tour from measure theory to random processes, via conditioning.Cambridge Series in Statistical and Probabilistic Mathematics, 13. Cambridge University Press, Cambridge, 2003. xvi+236 pp. ISBN: 0-521-82585-7 MR2016344 (2004m:60001)
  11. Fuchs, A.; Letta, G. Un résultat élémentaire de fiabilité. Application à la formule de Weierstrass sur la fonction gamma.(French) [An elementary reliability result. Application to the Weierstrass formula on the gamma function] Séminaire de Probabilités, XXV, 316--323, Lecture Notes in Math., 1485, Springer, Berlin, 1991. MR1187789 (93k:60034)
  12. Gordon, Louis. A stochastic approach to the gamma function. Amer. Math. Monthly 101 (1994), no. 9, 858--865. MR1300491 (95k:33003)
  13. Jacod. J, Kowalski. E and Nikeghbali. A: Mod-Gaussian convergence: new limit theorems in probability and number theory, http://arxiv.org/pdf/0807.4739 (2008).
  14. James, Lancelot F.; Roynette, Bernard; Yor, Marc. Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples. Probab. Surv. 5 (2008), 346--415. MR2476736
  15. Keating, J. P. $L$-functions and the characteristic polynomials of random matrices. Recent perspectives in random matrix theory and number theory, 251--277, London Math. Soc. Lecture Note Ser., 322, Cambridge Univ. Press, Cambridge, 2005. MR2166465 (2006m:11126)
  16. Keating, J. P.; Snaith, N. C. Random matrix theory and $zeta(1/2+it)$. Comm. Math. Phys. 214 (2000), no. 1, 57--89. MR1794265 (2002c:11107)
  17. Lebedev, N. N. Special functions and their applications.Revised edition, translated from the Russian and edited by Richard A. Silverman.Unabridged and corrected republication.Dover Publications, Inc., New York, 1972. xii+308 pp. MR0350075 (50 #2568)
  18. Recent perspectives in random matrix theory and number theory.Edited by F. Mezzadri and N. C. Snaith.London Mathematical Society Lecture Note Series, 322. Cambridge University Press, Cambridge, 2005. x+518 pp. ISBN: 978-0-521-62058-1; 0-521-62058-9 MR2145172 (2006c:11002)
  19. Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion.Third edition.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7 MR1725357 (2000h:60050)
  20. Sato. K: Lévy processes and infinitely divisible distributions}, Cambridge University Press 68, (1999).
  21. Thorin, Olof. On the infinite divisibility of the lognormal distribution. Scand. Actuar. J. 1977, no. 3, 121--148. MR0552135 (80m:60022)
  22. Voros, A. Spectral functions, special functions and the Selberg zeta function. Comm. Math. Phys. 110 (1987), no. 3, 439--465. MR0891947 (89b:58173)













Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | EJP

Electronic Communications in Probability. ISSN: 1083-589X