Couples de Wald indéfiniment divisibles. Exemples liés à la fonction gamma d'Euler et à la fonction zeta de Riemann

Roynette, Bernard; Yor, Marc

Annales de l'Institut Fourier, Tome 55 (2005) no. 4, pp. 1219-1283.

A toute mesure $c$ positive sur ${ℝ}_{+}$ telle que ${\int }_0^{\infty }(x\wedge x^2)c\left(dx\right)<\infty$, 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 $\lambda \ge 0,E\left(e^{\lambda X}\right)·E\left(e^{-\frac{{\lambda }^2}{2}H}\right)=1$. Plus généralement, à une mesure $c$ positive sur ${ℝ}_{+}$ telle que ${\int }_0^{\infty }{e}^{-\alpha x}{x}^{2}c\left(dx\right)<\infty$ pour tout $\alpha >{\alpha }_0$, 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.

To any positive measure $c$ on ${ℝ}_{+}$, such that :${\int }_0^{\infty }(x\wedge x^2)c\left(dx\right)<\infty$ we associate an infinitely divisible Wald couple, i.e. : a couple of random variables $(X,H)$ such that $X$ and $H$ are infinitely divisible, $H\ge 0$, and for any $\lambda \ge 0,E\left(e^{\lambda X}\right)·E\left(e^{-\frac{{\lambda }^2}{2}H}\right)=1$. More generally, to a positive measure $c$ on ${ℝ}_{+}$ which satisfies : ${\int }_0^{\infty }{e}^{-\alpha x}{x}^{2}c\left(dx\right)<\infty$ for every $\alpha >{\alpha }_0$, 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.

Classification : 60E67, 60E05, 60E10, 60G51
Mot clés : transformées de Laplace, lois indéfiniment divisibles, couples de Wald, fonctions gamma et zeta
Keywords: Laplace transforms, infinitely divisible laws, Wald couples, gamma and zeta functions

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