| Université Paris 6 Pierre et Marie Curie | Université Paris 7 Denis Diderot | |
| CNRS U.M.R. 7599 | ||
| ``Probabilités et Modèles Aléatoires'' | ||
Auteur(s):
Code(s) de Classification MSC:
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 : \begin{equation} \label{1} X \stackrel{(law)}{=} A +BX,\end{equation} 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