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, $EWαℓ(W)<∞$ if and only if $EZ1αℓ(Z1)<∞$; 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, $EWαℓ(W)<∞$ si et seulement si $EZ1αℓ(Z1)<∞$ ; Alsmeyer et Rösler (2004) [2] ont montré lʼéquivalence lorsque $α>1$ nʼest pas une puissance de 2. Nous le montrons ici pour tout $α>1$.

Accepted:
Published online:
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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