A stochastic fixed point equation for weighted minima and maxima
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 1, pp. 89-103.

Étant donné un ensemble fini ou dénombrable de nombres réel Tj, jJ, nous trouvons l'ensemble des solutions F de l'équation fonctionelle

W=dinfjJTjWj,
W et les Wj, jJ, sont des variables aléatoires mutuellement indépendantes ayant la loi F et =d signifie identité en loi. L'essentiel de ce travail concerne le cas où J2 et tous les Tj sont (strictement) positifs. Dans ce cas, toutes les solutions sont concentrées soit sur (0,) soit sur (-\infty ,0). Dans la situation la plus intéressante (et plus difficile) T a un exposant charactéristique α donné par jJTjα=1, et l'ensemble des solutions dépend du sous-groupe multiplicatif de >=(0,) généré par les Tj, qui est 1,> lui-même, ou r=rn:n pour quelque r>1. Le premier cas etant trivial, les points fixes non-triviaux dans le second cas sont ou bien les lois de Weibull ou bien leurs images réciproques sur (-\infty ,0) (si elles sont représentées par des variables aléatoires). Dans le troisième cas, il y a des solutions périodiques supplémentaires. Notre analyse est basée sur l'observation que le logarithme de la fonction de survie de chaque point fixe est harmonique relatif à Λ=j1δTj, c'est-à-dire Γ=ΓΛ, où dénote la convolution multiplicative. Cela nous permettrons l'utilisation du theorème puissant de Choquet et Deny.

Given any finite or countable collection of real numbers Tj, jJ, we find all solutions F to the stochastic fixed point equation

W=dinfjJTjWj,
where W and the Wj, jJ, are independent real-valued random variables with distribution F and =d means equality in distribution. The bulk of the necessary analysis is spent on the case when J2 and all Tj are (strictly) positive. Nontrivial solutions are then concentrated on either the positive or negative half line. In the most interesting (and difficult) situation T has a characteristic exponent α given by jJTjα=1 and the set of solutions depends on the closed multiplicative subgroup of >=(0,) generated by the Tj which is either 1,> itself or r=rn:n for some r>1. The first case being trivial, the nontrivial fixed points in the second case are either Weibull distributions or their reciprocal reflections to the negative half line (when represented by random variables), while in the third case further periodic solutions arise. Our analysis builds on the observation that the logarithmic survival function of any fixed point is harmonic with respect to Λ=j1δTj, i.e. Γ=ΓΛ, where means multiplicative convolution. This will enable us to apply the powerful Choquet-Deny theorem.

DOI : 10.1214/07-AIHP104
Classification : 60E05, 60J80
Mots-clés : stochastic fixed point equation, weighted minima and maxima, weighted branching process, harmonic analysis on trees, Choquet-Deny theorem, Weibull distributions
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Alsmeyer, Gerold; Rösler, Uwe. A stochastic fixed point equation for weighted minima and maxima. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 1, pp. 89-103. doi : 10.1214/07-AIHP104. https://www.numdam.org/articles/10.1214/07-AIHP104/

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