Weighted moments for the limit of a normalized supercritical Galton–Watson process

Liang, Xingang; Liu, Quansheng

doi:10.1016/j.crma.2013.09.015

Probability theory

Weighted moments for the limit of a normalized supercritical Galton–Watson process
[Moments pondérés pour la limite dʼun processus de Galton–Watson normalisé supercritique]

Présenté par : Jean-Pierre Kahane

Liang, Xingang ^1,² ; Liu, Quansheng ^2,³

¹ Beijing Technology and Business Univ., School of Science, 100048 Beijing, China
² Université de Bretagne-Sud, CNRS UMR 6205, LMBA, campus de Tohannic, 56000 Vannes, France
³ Changsha Univ. of Science and Technology, School of Mathematics and Computing Science, 410004 Changsha, China

Comptes Rendus. Mathématique, Tome 351 (2013) no. 19-20, pp. 769-773.

Résumé
Abstract

Soient $(Z_{n})$ un processus de Galton–Watson surcritique et W la limite de la population normalisée $Z_{n} / m^{n}$ , où $m = E Z_{1} > 1$ est la moyenne de la loi de reproduction. Soit ℓ une fonction positive à variation lente en ∞. Bingham et Doney (1974) [4] ont montré que, pour $α > 0$ non entier, $E W^{α} ℓ (W) < \infty$ si et seulement si $E Z_{1}^{α} ℓ (Z_{1}) < \infty$ ; Alsmeyer et Rösler (2004) [2] ont montré lʼéquivalence lorsque $α > 1$ nʼest pas une puissance de 2. Nous le montrons ici pour tout $α > 1$ .

Let $(Z_{n})$ be a supercritical Galton–Watson process, and let W be the limit of the normalized population size $Z_{n} / m^{n}$ , where $m = E Z_{1} > 1$ is the mean of the offspring distribution. Let ℓ be a positive function slowly varying at ∞. Bingham and Doney (1974) [4] showed that for $α > 1$ not an integer, $E W^{α} ℓ (W) < \infty$ if and only if $E Z_{1}^{α} ℓ (Z_{1}) < \infty$ ; Alsmeyer and Rösler (2004) [2] proved the equivalence for $α > 1$ not a dyadic power. Here we prove it for all $α > 1$ .

Reçu le : 2013-07-14
Accepté le : 2013-09-18
Publié le : 2013-10-01

DOI : 10.1016/j.crma.2013.09.015

Affiliations des auteurs :

Liang, Xingang ^{1, 2} ; Liu, Quansheng ^{2, 3}

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     author = {Liang, Xingang and Liu, Quansheng},
     title = {Weighted moments for the limit of a normalized supercritical {Galton{\textendash}Watson} process},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {769--773},
     publisher = {Elsevier},
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     doi = {10.1016/j.crma.2013.09.015},
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Liang, Xingang; Liu, Quansheng. Weighted moments for the limit of a normalized supercritical Galton–Watson process. Comptes Rendus. Mathématique, Tome 351 (2013) no. 19-20, pp. 769-773. doi : 10.1016/j.crma.2013.09.015. http://www.numdam.org/articles/10.1016/j.crma.2013.09.015/

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