Comptes Rendus
Probability Theory
Tail behavior of laws stable by random weighted mean
[Variation régulière des lois stables par moyenne pondérée aléatoire]
Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 347-352.

Soit (N,A1,A2,) une suite de variables aléatoires avec NN{} et AiR+. Nous nous sommes intéressés aux propriétés asymptotiques des solutions de lʼéquation en distribution Z=i=1NAiZi, où Zi sont des variables aléatoires non-négatives, mutuellement indépendantes et indépendantes de (N,A1,A2,), chacune a la même loi que Z qui est inconnue. Pour une solution Z0 de moyenne finie, nous montrons que sous une condition de moment naturelle, la variation régulière de la probabilité de queue P(Z>x) (x) est équivalente à celle de P(Y1>x), où Y1=i=1NAi. 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,A1,A2,) be a sequence of random variables with NN{} and AiR+. We are interested in asymptotic properties of solutions of the distributional equation Z=i=1NAiZi, where Zi are nonnegative random variables independent of each other and independent of (N,A1,A2,), each has the same distribution as Z which is unknown. For a solution Z0 with finite mean, we show that under a natural moment condition, the regular variation of P(Z>x) (x) is equivalent to that of P(Y1>x), where Y1=i=1NAi. 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 :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.01.029

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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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/

[1] N.H. Bingham; R.A. Doney Asymptotic properties of supercritical branching processes I: The Galton–Watson processes, Adv. in Appl. Probab., Volume 6 (1974), pp. 711-731

[2] N.H. Bingham; R.A. Doney Asymptotic properties of supercritical branching processes II: Crump–Mode and Jirina processes, Adv. in Appl. Probab., Volume 7 (1975), pp. 66-82

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