Harmonic moments and large deviations for a supercritical branching process in a random environment

Chunmao Huang; Quansheng Liu

doi:10.1016/j.crma.2011.10.001

Probability Theory

Harmonic moments and large deviations for a supercritical branching process in a random environment
[Moments harmoniques et grandes déviations pour un processus de branchement sur-critique dans un environnement aléatoire]

Présenté par : the Editorial Board

Chunmao Huang ^1,² ; Quansheng Liu ^1,²

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

Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1199-1202.

Résumé
Abstract

Soient $(Z_{n})$ un processus de branchement sur-critique dans un environnement aléatoire ξ, et W la limite de la population normalisée $Z_{n} / E [Z_{n} | ξ]$ . Nous montrons les principes de grande déviation et de déviation modérée pour la suite $\log Z_{n}$ en trouvant un équivalent des moments de $Z_{n}$ et un critère pour lʼexistence des moments harmoniques de W.

Let $(Z_{n})$ be a supercritical branching process in a random environment ξ, and W be the limit of the normalized population size $Z_{n} / E [Z_{n} | ξ]$ . We show large and moderate deviation principles for the sequence $\log Z_{n}$ (with appropriate normalization) by finding an equivalence of the moments of $Z_{n}$ and a criterion for the existence of harmonic moments of W.

Reçu le : 2011-07-29
Accepté le : 2011-10-03
Publié le : 2011-10-22

DOI : 10.1016/j.crma.2011.10.001

Affiliations des auteurs :

Chunmao Huang ^{1, 2} ; Quansheng Liu ^{1, 2}

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

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     author = {Chunmao Huang and Quansheng Liu},
     title = {Harmonic moments and large deviations for a supercritical branching process in a random environment},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1199--1202},
     publisher = {Elsevier},
     volume = {349},
     number = {21-22},
     year = {2011},
     doi = {10.1016/j.crma.2011.10.001},
     language = {en},
}

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Chunmao Huang; Quansheng Liu. Harmonic moments and large deviations for a supercritical branching process in a random environment. Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1199-1202. doi : 10.1016/j.crma.2011.10.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.10.001/

Bibliographie
Cité par

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[3] Y. Guivarcʼh; Q. Liu Propriétés asymptotiques des processus de branchement en environnement aléatoire, C. R. Acad. Sci. Paris, Ser. I, Volume 332 (2001), pp. 339-344

[4] B. Hambly On the limit distribution of a supercritical branching process in a random environment, J. Appl. Probab., Volume 29 (1992), pp. 499-518

[5] Q. Liu Asymptotic properties and absolute continuity of laws stable by random weighted mean, Stochastic Process. Appl., Volume 95 (2001), pp. 83-107

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[7] D. Tanny Limit theorems for branching processes in a random environment, Ann. Probab., Volume 5 (1977), pp. 100-116

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