Tail behavior of laws stable by random weighted mean

Xingang Liang; Quansheng Liu

doi:10.1016/j.crma.2011.01.029

Probability Theory

Tail behavior of laws stable by random weighted mean
[Variation régulière des lois stables par moyenne pondérée aléatoire]

Présenté par : the Editorial Board

Xingang Liang ^1,² ; Quansheng Liu ^1,²

¹ LMAM, Université de Bretagne-Sud, Campus de Tohannic, BP 573, 56017 Vannes, France
² Université Européenne de Bretagne, France

Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 347-352.

Résumé
Abstract

Soit $(N, A_{1}, A_{2}, \dots)$ une suite de variables aléatoires avec $N \in N \cup {\infty}$ et $A_{i} \in R_{+}$ . Nous nous sommes intéressés aux propriétés asymptotiques des solutions de lʼéquation en distribution $Z = \sum_{i = 1}^{N} A_{i} Z_{i}$ , où $Z_{i}$ sont des variables aléatoires non-négatives, mutuellement indépendantes et indépendantes de $(N, A_{1}, A_{2}, \dots)$ , chacune a la même loi que Z qui est inconnue. Pour une solution $Z ⩾ 0$ de moyenne finie, nous montrons que sous une condition de moment naturelle, la variation régulière de la probabilité de queue $P (Z > x)$ $(x \to \infty)$ est équivalente à celle de $P (Y_{1} > x)$ , où $Y_{1} = \sum_{i = 1}^{N} A_{i}$ . Les résultats généralisent les théorèmes correspondants de Bingham et Doney (1974, 1975) [1,2] et de Meyer (1982) [6] sur les processus de Galton–Watson et de Crump–Mode–Jirina, et améliorent ceux dʼIksanov et Polotskiy (2006) [7] sur les marches aléatoires branchantes.

Let $(N, A_{1}, A_{2}, \dots)$ be a sequence of random variables with $N \in N \cup {\infty}$ and $A_{i} \in R_{+}$ . We are interested in asymptotic properties of solutions of the distributional equation $Z = \sum_{i = 1}^{N} A_{i} Z_{i}$ , where $Z_{i}$ are nonnegative random variables independent of each other and independent of $(N, A_{1}, A_{2}, \dots)$ , each has the same distribution as Z which is unknown. For a solution $Z ⩾ 0$ with finite mean, we show that under a natural moment condition, the regular variation of $P (Z > x)$ $(x \to \infty)$ is equivalent to that of $P (Y_{1} > x)$ , where $Y_{1} = \sum_{i = 1}^{N} A_{i}$ . The results generalize the corresponding theorems of Bingham and Doney (1974, 1975) [1,2] and de Meyer (1982) [6] on Galton–Watson processes and Crump–Mode–Jirina processes, and improve those of Iksanov and Polotskiy (2006) [7] on branching random walks.

Reçu le : 2011-01-13
Accepté le : 2011-01-31
Publié le : 2011-02-18

DOI : 10.1016/j.crma.2011.01.029

Affiliations des auteurs :

Xingang Liang ^{1, 2} ; Quansheng Liu ^{1, 2}

¹ LMAM, Université de Bretagne-Sud, Campus de Tohannic, BP 573, 56017 Vannes, France
² Université Européenne de Bretagne, France

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     title = {Tail behavior of laws stable by random weighted mean},
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Xingang Liang; Quansheng Liu. Tail behavior of laws stable by random weighted mean. Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 347-352. doi : 10.1016/j.crma.2011.01.029. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.01.029/

Bibliographie
Cité par

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Xingang Liang; Quansheng Liu Regular variation of fixed points of the smoothing transform, Stochastic Processes and their Applications, Volume 130 (2020) no. 7, p. 4104 | DOI:10.1016/j.spa.2019.11.011
Alexander Iksanov; Konrad Kolesko; Matthias Meiners Stable-like fluctuations of Biggins’ martingales, Stochastic Processes and their Applications, Volume 129 (2019) no. 11, p. 4480 | DOI:10.1016/j.spa.2018.11.022

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