## The barnes G function and its relations with sums and products of generalized Gamma variables, in preparation

Citations: | 3 - 2 self |

### BibTeX

@MISC{Nikeghbali_thebarnes,

author = {A. Nikeghbali and Marc Yor},

title = {The barnes G function and its relations with sums and products of generalized Gamma variables, in preparation},

year = {}

}

### OpenURL

### Abstract

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, and that the reciprocal of the Barnes G-function has a Lévy-Khintchin type representation. These results lead us to introduce the so called generalized gamma convolution variables.

### Citations

1057 |
Continuous martingales and brownian motion
- Revuz, Yor
- 1991
(Show Context)
Citation Context ...Bessel processes. Let (R2n (t)) t≥0 denote a BES(2n) process, starting from 0, with dimension 2n; we need to consider the sequence (R2n) n=1,2,... of such independent processes. It is well known (see =-=[17]-=- for example) that: Moreover, we have: for fixed t > 0, R 2 2n (t) law = (2t) γn. for fixed t > 0, R 2 ∫ t 2n (t) = 2 0 √ R2 2n (s)dβ(n) s + 2nt the stochastic differential equation of R2 2n , driven ... |

797 |
Lévy process and Infinitely Divisible Distributions
- Sato
- 1999
(Show Context)
Citation Context ...asure on R+, called the Thorin measure of Y . Remark 3.2. Y is a selfdecomposable random variable because its Lévy measure can be written as ν (dx) = dx h(x) with h a decreasing function (see, x e.g. =-=[18]-=-, p.95). Remark 3.3. We shall require Y to have finite first and second moments; these moments can be easily computed with the help of the Thorin measure µ (dξ): ∫ E [Y ] = µ−1 = σ 2 = E [ Y 2] − (E [... |

608 | Special Functions - Andrews, Askey, et al. - 1990 |

76 |
Generalized Gamma Convolutions and Related Classes of Distributions and Densities
- Bondesson
- 1992
(Show Context)
Citation Context ...o Y . We shall further assume that ∫ µ (dξ) 1 < ∞, ξ2 which, as we shall see is equivalent to the existence of a second moment for Y . The GGC variables have been studied by Thorin [19] and Bondesson =-=[6]-=-, see, e.g., [12] for a recent survey of this topic. Theorem 1.5. Let Y be a GGC variable, and let (SN) as in (1.13). We note σ 2 = E [ Y 2] − (E [Y ]) 2 . Then the following limit theorem for (SN) ho... |

72 | On the distribution and asymptotic results for exponential functionals of Lévy processes,” in Exponential functionals and principal values related to Brownian motion
- Carmona, Petit, et al.
- 1997
(Show Context)
Citation Context ...ψ (a) ≡ Γ′ (a) Γ (a) . □10 A. NIKEGHBALI AND MARC YOR Proof. This is classical: it follows from (1.7) and some integral representation of the ψ-function, see, e.g. Lebedev [15] and Carmona-Petit-Yor =-=[7]-=- where this lemma is also used. □ We are now in a position to prove Theorem 1.4. We start by proving the first part, i.e. formula (1.12). Let us write IN (λ) := 1 Nλ2 /2 E ⎡⎛ ⎞λ N∏ ⎢ ⎣⎝ γj⎠ ⎤ ⎥ ⎦ . Th... |

61 | Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions
- Biane, Pitman, et al.
- 2001
(Show Context)
Citation Context ... by computing its density (for an example of occurrence of the random variable Q in the theory of stochastic processes and some relation with the theory of the Riemann Zeta function, see e.g. [5] and =-=[4]-=-). From the definition of Q via its Mellin transform, we get for f : R+ → R+: E [f (Q)] = 3 π 2 ∞∑ n=1 ( 1 n 2 ∫ ∞ dt t 2 exp (−t)f 0 ( t n ))BARNES G-FUNCTION AND GAMMA VARIABLES 9 Hence by some ele... |

60 |
Valeurs principales associées aux temps locaux browniens
- Biane, Yor
- 1987
(Show Context)
Citation Context ...riable Q by computing its density (for an example of occurrence of the random variable Q in the theory of stochastic processes and some relation with the theory of the Riemann Zeta function, see e.g. =-=[5]-=- and [4]). From the definition of Q via its Mellin transform, we get for f : R+ → R+: E [f (Q)] = 3 π 2 ∞∑ n=1 ( 1 n 2 ∫ ∞ dt t 2 exp (−t)f 0 ( t n ))BARNES G-FUNCTION AND GAMMA VARIABLES 9 Hence by ... |

47 | Spectral functions, special functions and the Selberg zeta function - Voros - 1987 |

18 |
Exercises in probability. A guided tour from measure theory to random processes, via conditioning
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- 2003
(Show Context)
Citation Context ...ting-Snaith [14]. Formula (1.11) follows from (1.10) once one recalls formula (1.7). 2.3. The characteristic polynomial and beta variables. One can use the beta-gamma algebra (see, e.g., Chaumont-Yor =-=[8]-=-, p.93-98, and the references therein) and (1.11) to represent (in law) the characteristic polynomial as products of beta variables. More precisely, Theorem 2.1. With the notations of Theorem 1.3, we ... |

17 |
The theory of G-function, Quart
- Barnes
(Show Context)
Citation Context .... These results lead us to introduce the so called generalized gamma convolution variables. 1. Introduction, motivation and main results The Barnes G-function, which was first introduced by Barnes in =-=[3]-=- (see also [1]), may be defined via its infinite product representation: G(1 + z) = (2π) z/2 [ exp − 1 [ ] 2 (1 + γ) z + z 2 ] ∏∞ ( 1 + n=1 z ) n exp n [ −z + z2 ] 2n (1.1) where γ is the Euler consta... |

17 |
A stochastic approach to the gamma function
- Gordon
- 1994
(Show Context)
Citation Context ...t (1.1) in terms of a limiting distribution involving gamma variables (we note that, concerning the Gamma function, similar translationsBARNES G-FUNCTION AND GAMMA VARIABLES 3 have been presented in =-=[10]-=- and [9]). Let us recall that a gamma variable γa with parameter a > 0 is distributed as: and has Laplace transform and Mellin transform: P (γa ∈ dt) = ta−1 exp (−t)dt , t > 0 (1.5) Γ (a) E [exp (−λγa... |

17 |
Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples. arXiv:0708.3932V1 [math.PR] 29
- James, Roynette, et al.
- 2007
(Show Context)
Citation Context ...urther assume that ∫ µ (dξ) 1 < ∞, ξ2 which, as we shall see is equivalent to the existence of a second moment for Y . The GGC variables have been studied by Thorin [19] and Bondesson [6], see, e.g., =-=[12]-=- for a recent survey of this topic. Theorem 1.5. Let Y be a GGC variable, and let (SN) as in (1.13). We note σ 2 = E [ Y 2] − (E [Y ]) 2 . Then the following limit theorem for (SN) holds: if λ > 0, 1 ... |

7 |
Lebedev Special Functions and their Application
- N
- 1971
(Show Context)
Citation Context ...1 + λu) du u(1 − exp (−u)) ψ (a) ≡ Γ′ (a) Γ (a) . □10 A. NIKEGHBALI AND MARC YOR Proof. This is classical: it follows from (1.7) and some integral representation of the ψ-function, see, e.g. Lebedev =-=[15]-=- and Carmona-Petit-Yor [7] where this lemma is also used. □ We are now in a position to prove Theorem 1.4. We start by proving the first part, i.e. formula (1.12). Let us write IN (λ) := 1 Nλ2 /2 E ⎡⎛... |

6 |
NC Snaith ”Random Matrix Theory and ζ(1/2+it
- Keating
(Show Context)
Citation Context ... = z 1 (log (2π) − 1) − 2 2 (1 + γ) z2 + ∞∑ n=3 (−1) n−1 ζ (n − 1) zn n (1.2) where ζ denotes the Riemann zeta function. This Barnes G-function has recently occurred in the work of Keating and Snaith =-=[14]-=- in their celebrated moments conjecture for the Riemann zeta Date: February 1, 2008. 2000 Mathematics Subject Classification. 60F99, 60E07, 60E10. Key words and phrases. Barnes G-function, beta-gamma ... |

5 |
On the infinite divisibility of the lognormal distribution, Scand
- Thorin
- 1977
(Show Context)
Citation Context ...easure associated to Y . We shall further assume that ∫ µ (dξ) 1 < ∞, ξ2 which, as we shall see is equivalent to the existence of a second moment for Y . The GGC variables have been studied by Thorin =-=[19]-=- and Bondesson [6], see, e.g., [12] for a recent survey of this topic. Theorem 1.5. Let Y be a GGC variable, and let (SN) as in (1.13). We note σ 2 = E [ Y 2] − (E [Y ]) 2 . Then the following limit t... |

2 | L-functions and the characteristic polynomials of random matrices - Keating - 2005 |

1 |
Contribution to the theory of the Barnes function, preprint
- Adamchik
- 2001
(Show Context)
Citation Context ...s lead us to introduce the so called generalized gamma convolution variables. 1. Introduction, motivation and main results The Barnes G-function, which was first introduced by Barnes in [3] (see also =-=[1]-=-), may be defined via its infinite product representation: G(1 + z) = (2π) z/2 [ exp − 1 [ ] 2 (1 + γ) z + z 2 ] ∏∞ ( 1 + n=1 z ) n exp n [ −z + z2 ] 2n (1.1) where γ is the Euler constant. From (1.1)... |

1 |
Un rsultat élémentaire de fiabilité. Application la formule de Weierstrass sur la fonction gamma
- Fuchs, Letta
- 1991
(Show Context)
Citation Context ...n terms of a limiting distribution involving gamma variables (we note that, concerning the Gamma function, similar translationsBARNES G-FUNCTION AND GAMMA VARIABLES 3 have been presented in [10] and =-=[9]-=-). Let us recall that a gamma variable γa with parameter a > 0 is distributed as: and has Laplace transform and Mellin transform: P (γa ∈ dt) = ta−1 exp (−t)dt , t > 0 (1.5) Γ (a) E [exp (−λγa)] = E [... |