Probability theory
Weighted moments for the limit of a normalized supercritical Galton–Watson process
Comptes Rendus. Mathématique, Volume 351 (2013) no. 19-20, pp. 769-773.

Let (Zn) be a supercritical Galton–Watson process, and let W be the limit of the normalized population size Zn/mn, where m=EZ1>1 is the mean of the offspring distribution. Let be a positive function slowly varying at ∞. Bingham and Doney (1974) [4] showed that for α>1 not an integer, EWα(W)< if and only if EZ1α(Z1)<; Alsmeyer and Rösler (2004) [2] proved the equivalence for α>1 not a dyadic power. Here we prove it for all α>1.

Soient (Zn) un processus de Galton–Watson surcritique et W la limite de la population normalisée Zn/mn, où m=EZ1>1 est la moyenne de la loi de reproduction. Soit une fonction positive à variation lente en ∞. Bingham et Doney (1974) [4] ont montré que, pour α>0 non entier, EWα(W)< si et seulement si EZ1α(Z1)< ; 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.

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DOI: 10.1016/j.crma.2013.09.015
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
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Liang, Xingang; Liu, Quansheng. Weighted moments for the limit of a normalized supercritical Galton–Watson process. Comptes Rendus. Mathématique, Volume 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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