Electronic Communications in Probability

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

Ashkan Nikeghbali and Marc Yor

Full-text: Open access

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


Export citation

References

  • 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.
  • 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
  • Barnes,E.W: The theory of the G-function, Quart. J. Pure Appl. Math. 31 (1899), 264–314.
  • 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).
  • 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.
  • 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
  • Bottcher, A. ; Silbermann. B: Analysis of Toeplitz operators, Springer-Verlag, Berlin. Second Edition (2006).
  • 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.
  • 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.
  • 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
  • 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.
  • Gordon, Louis. A stochastic approach to the gamma function. Amer. Math. Monthly 101 (1994), no. 9, 858–865.
  • 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).
  • James, Lancelot F.; Roynette, Bernard; Yor, Marc. Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples. Probab. Surv. 5 (2008), 346–415.
  • 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.
  • Keating, J. P.; Snaith, N. C. Random matrix theory and $zeta(1/2+it)$. Comm. Math. Phys. 214 (2000), no. 1, 57–89.
  • 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.
  • 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
  • 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
  • Sato. K: Lévy processes and infinitely divisible distributions}, Cambridge University Press 68, (1999).
  • Thorin, Olof. On the infinite divisibility of the lognormal distribution. Scand. Actuar. J. 1977, no. 3, 121–148.
  • Voros, A. Spectral functions, special functions and the Selberg zeta function. Comm. Math. Phys. 110 (1987), no. 3, 439–465.