Couples de Wald indéfiniment divisibles. Exemples liés à la fonction gamma d'Euler et à la fonction zeta de Riemann  [ Infinitely divisible Wald's couples. Examples linked with the Euler gamma and the Riemann zeta functions ]
Annales de l'Institut Fourier, Volume 55 (2005) no. 4, p. 1219-1283

To any positive measure c on + , such that : 0 (xx 2 )c(dx)< we associate an infinitely divisible Wald couple, i.e. : a couple of random variables (X,H) such that X and H are infinitely divisible, H0, and for any λ0,Ee λX ·Ee -λ 2 2H =1. More generally, to a positive measure c on + which satisfies : 0 e -αx x 2 c(dx)< 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.

A toute mesure c positive sur + telle que 0 (xx 2 )c(dx)<, 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, H0, et pour tout λ0,Ee λX ·Ee -λ 2 2H =1. Plus généralement, à une mesure c positive sur + telle que 0 e -αx x 2 c(dx)< pour tout α>α 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.

DOI : https://doi.org/10.5802/aif.2125
Classification:  60E67,  60E05,  60E10,  60G51
Keywords: Laplace transforms, infinitely divisible laws, Wald couples, gamma and zeta functions
@article{AIF_2005__55_4_1219_0,
     author = {Roynette, Bernard and Yor, Marc},
     title = {Couples de Wald ind\'efiniment divisibles. Exemples li\'es \`a la fonction gamma d'Euler et \`a la fonction zeta de Riemann},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {4},
     year = {2005},
     pages = {1219-1283},
     doi = {10.5802/aif.2125},
     zbl = {1083.60012},
     mrnumber = {2157168},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2005__55_4_1219_0}
}
Roynette, Bernard; 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, Volume 55 (2005) no. 4, pp. 1219-1283. doi : 10.5802/aif.2125. http://www.numdam.org/item/AIF_2005__55_4_1219_0/

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