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

Given any finite or countable collection of real numbers ${T}_{j}$, $j\in J$, we find all solutions $F$ to the stochastic fixed point equation $W\stackrel{\mathrm{d}}{=}\underset{j\in J}{inf}{T}_{j}{W}_{j},$ where $W$ and the ${W}_{j}$, $j\in J$, are independent real-valued random variables with distribution $F$ and $\stackrel{\mathrm{d}}{=}$ means equality in distribution. The bulk of the necessary analysis is spent on the case when $\left|J\right|\ge 2$ and all ${T}_{j}$ 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 $\alpha$ given by ${\sum }_{j\in J}{T}_{j}^{\alpha }=1$ and the set of solutions depends on the closed multiplicative subgroup of ${ℝ}^{>}=\left(0,\infty \right)$ generated by the ${T}_{j}$ which is either $\left\{1\right\},{ℝ}^{>}$ itself or ${r}^{ℤ}=\left\{{r}^{n}:n\in ℤ\right\}$ 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 $\Lambda ={\sum }_{j\ge 1}{\delta }_{{T}_{j}}$, i.e. $\Gamma =\Gamma ☆\Lambda$, where $☆$ means multiplicative convolution. This will enable us to apply the powerful Choquet-Deny theorem.

Étant donné un ensemble fini ou dénombrable de nombres réel ${T}_{j}$, $j\in J$, nous trouvons l'ensemble des solutions $F$ de l'équation fonctionelle $W\stackrel{\mathrm{d}}{=}\underset{j\in J}{inf}{T}_{j}{W}_{j},$$W$ et les ${W}_{j}$, $j\in J$, sont des variables aléatoires mutuellement indépendantes ayant la loi $F$ et $\stackrel{\mathrm{d}}{=}$ signifie identité en loi. L'essentiel de ce travail concerne le cas où $\left|J\right|\ge 2$ et tous les ${T}_{j}$ sont (strictement) positifs. Dans ce cas, toutes les solutions sont concentrées soit sur $\left(0,\infty \right)$ soit sur (-\infty ,0). Dans la situation la plus intéressante (et plus difficile) $T$ a un exposant charactéristique $\alpha$ donné par ${\sum }_{j\in J}{T}_{j}^{\alpha }=1$, et l'ensemble des solutions dépend du sous-groupe multiplicatif de ${ℝ}^{>}=\left(0,\infty \right)$ généré par les ${T}_{j}$, qui est $\left\{1\right\},{ℝ}^{>}$ lui-même, ou ${r}^{ℤ}=\left\{{r}^{n}:n\in ℤ\right\}$ 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 $\Lambda ={\sum }_{j\ge 1}{\delta }_{{T}_{j}}$, c'est-à-dire $\Gamma =\Gamma ☆\Lambda$, où $☆$ dénote la convolution multiplicative. Cela nous permettrons l'utilisation du theorème puissant de Choquet et Deny.

DOI : https://doi.org/10.1214/07-AIHP104
Classification:  60E05,  60J80
Keywords: stochastic fixed point equation, weighted minima and maxima, weighted branching process, harmonic analysis on trees, Choquet-Deny theorem, Weibull distributions
@article{AIHPB_2008__44_1_89_0,
author = {Alsmeyer, Gerold and R\"osler, Uwe},
title = {A stochastic fixed point equation for weighted minima and maxima},
journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
publisher = {Gauthier-Villars},
volume = {44},
number = {1},
year = {2008},
pages = {89-103},
doi = {10.1214/07-AIHP104},
zbl = {1176.60006},
mrnumber = {2451572},
language = {en},
url = {http://www.numdam.org/item/AIHPB_2008__44_1_89_0}
}

Alsmeyer, Gerold; Rösler, Uwe. A stochastic fixed point equation for weighted minima and maxima. Annales de l'I.H.P. Probabilités et statistiques, Volume 44 (2008) no. 1, pp. 89-103. doi : 10.1214/07-AIHP104. http://www.numdam.org/item/AIHPB_2008__44_1_89_0/

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