Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

Affine random equations and the stable (1/2) distribution

Auteur(s):

Code(s) de Classification MSC:

• 60E07 Infinitely divisible distributions; stable distributions

Résumé: In various domains in probabilistic studies, affine random equations are stu\-died, i.e. given a pair $(A,B)$ of random variables, one is interested in the study of all possible variables $X$ such that : $$\label{1} X \stackrel{(law)}{=} A +BX,$$ where, on the right-hand side, $X$ is independent from the pair $(A,B)$. (See, e.g., Babillot, Bougerol and Elie \cite{B-B-E} for some recent study in the so-called critical case, and the references therein).\\ Converse studies, for which the law of $X$ is given a priori, and one looks for all possible pairs $(A,B)$ of random variables satisfying (\ref{1}) seem to be less popular. In the present note, we discuss the important particular case of such a converse study when $X \equiv T$ is the stable $(\frac{1}{2})$ variable, i.e. : $P(T \in dt) = \frac{dt}{\sqrt{2\pi t^3}} \exp(-\frac{1}{2t}).$

Mots Clés: stable 1/2 distribution; affine random equations;

Date: 1999-03-10

Prépublication numéro: PMA-489