Infinitely divisible Wald's couples. Examples linked with the Euler gamma and the Riemann zeta functions

Bernard Roynette[1]; Marc Yor

  • [1] Institut Elie Cartan, département de Mathématiques, B.P. 239, 54506 Vandoeuvre Les Nancy Cedex (France), Université Paris VI, Laboratoire de Probabilités et Modèles Aléatoires, Tour 56 - 3ème étage, 75252 Paris Cedex 05 (France)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 4, page 1219-1283
  • ISSN: 0373-0956

Abstract

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To any positive measure c on ℝ + , such that : ∫ 0 ∞ ( x ∧ x 2 ) c ( d x ) < ∞ we associate an infinitely divisible Wald couple, i.e. : a couple of random variables ( X , H ) such that X and H are infinitely divisible, H ≥ 0 , and for any λ ≥ 0 , E ( e λ X ) · E ( e - λ 2 2 H ) = 1 . More generally, to a positive measure c on ℝ + which satisfies : ∫ 0 ∞ e - α x x 2 c ( d x ) < ∞ for every α > α 0 , we associate an “Esscher family” of infinitely divisible Wald couples. We give many examples of such Esscher families and we prove that the particular ones which are associated with the gamma and the zeta functions enjoy remarkable properties.

How to cite

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Roynette, Bernard, and Yor, Marc. "Couples de Wald indéfiniment divisibles. Exemples liés à la fonction gamma d'Euler et à la fonction zeta de Riemann." Annales de l’institut Fourier 55.4 (2005): 1219-1283. <http://eudml.org/doc/116218>.

@article{Roynette2005,
abstract = {A toute mesure $c$ positive sur $\{\mathbb \{R\}_+\}$ telle que $\int ^\{\infty \}_0 (x \wedge x^2) c(dx) &lt; \{\infty \}$, nous associons un couple de Wald indéfiniment divisible, i.e. un couple de variables aléatoires $(X,H)$ tel que $X$ et $H$ sont indéfiniment divisibles, $H \ge 0$, et pour tout $\lambda \{\ge \} 0, E \big (e^\{\lambda X\} \big ) \cdot E \big (e^\{- \{\lambda ^2 \over 2\} H\}\big )=1$. Plus généralement, à une mesure $c$ positive sur $\{\mathbb \{R\}_+\}$ telle que $\int _0^\{\infty \}e^\{- \alpha x\} x^2 \; c(dx) &lt; \infty $ pour tout $\alpha &gt; \alpha _0$, nous associons une “famille d’Esscher” de couples de Wald indéfiniment divisibles. Nous donnons de nombreux exemples de telles familles d’Esscher. Celles liées à la fonction gamma et à la fonction zeta de Riemann possèdent des propriétés remarquables.},
affiliation = {Institut Elie Cartan, département de Mathématiques, B.P. 239, 54506 Vandoeuvre Les Nancy Cedex (France), Université Paris VI, Laboratoire de Probabilités et Modèles Aléatoires, Tour 56 - 3ème étage, 75252 Paris Cedex 05 (France)},
author = {Roynette, Bernard, Yor, Marc},
journal = {Annales de l’institut Fourier},
keywords = {Laplace transforms; infinitely divisible laws; Wald couples; gamma and zeta functions},
language = {fre},
number = {4},
pages = {1219-1283},
publisher = {Association des Annales de l'Institut Fourier},
title = {Couples de Wald indéfiniment divisibles. Exemples liés à la fonction gamma d'Euler et à la fonction zeta de Riemann},
url = {http://eudml.org/doc/116218},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Roynette, Bernard
AU - Yor, Marc
TI - Couples de Wald indéfiniment divisibles. Exemples liés à la fonction gamma d'Euler et à la fonction zeta de Riemann
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 4
SP - 1219
EP - 1283
AB - A toute mesure $c$ positive sur ${\mathbb {R}_+}$ telle que $\int ^{\infty }_0 (x \wedge x^2) c(dx) &lt; {\infty }$, nous associons un couple de Wald indéfiniment divisible, i.e. un couple de variables aléatoires $(X,H)$ tel que $X$ et $H$ sont indéfiniment divisibles, $H \ge 0$, et pour tout $\lambda {\ge } 0, E \big (e^{\lambda X} \big ) \cdot E \big (e^{- {\lambda ^2 \over 2} H}\big )=1$. Plus généralement, à une mesure $c$ positive sur ${\mathbb {R}_+}$ telle que $\int _0^{\infty }e^{- \alpha x} x^2 \; c(dx) &lt; \infty $ pour tout $\alpha &gt; \alpha _0$, nous associons une “famille d’Esscher” de couples de Wald indéfiniment divisibles. Nous donnons de nombreux exemples de telles familles d’Esscher. Celles liées à la fonction gamma et à la fonction zeta de Riemann possèdent des propriétés remarquables.
LA - fre
KW - Laplace transforms; infinitely divisible laws; Wald couples; gamma and zeta functions
UR - http://eudml.org/doc/116218
ER -

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