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

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.09.015
Xingang Liang 1, 2 ; Quansheng Liu 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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     title = {Weighted moments for the limit of a normalized supercritical {Galton{\textendash}Watson} process},
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     pages = {769--773},
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Xingang Liang; Quansheng Liu. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.09.015/

[1] G. Alsmeyer; D. Kuhlbusch Double martingale structure and existence of ϕ-moments for weighted branching processes, Münster J. Math., Volume 3 (2010), pp. 163-211

[2] G. Alsmeyer; U. Rösler On the existence of ϕ-moments of the limit of a normalized supercritical Galton–Watson process, J. Theor. Probab., Volume 17 (2004) no. 4, pp. 905-928

[3] K.B. Athreya; P.E. Ney Branching Processes, Springer, New York, 1972

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

[5] N.H. Bingham; C.M. Goldie; J.L. Teugels Regular Variation, Cambridge Univ. Press, Cambridge, 1987

[6] Y. Chow; H. Teicher Probability Theory: Independence, Interchangeability, Martingales, Springer-Verlag, 1995

[7] R. Durrett; T. Liggett Fixed points of the smoothing transformation, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 64 (1983), pp. 275-301

[8] T.E. Harris The Theory of Branching Processes, Springer, Berlin, 1963

[9] A.M. Iksanov Elementary fixed points of the BRW smoothing transforms with infinite number of summands, Stoch. Process. Appl., Volume 114 (2004) no. 1, pp. 27-50

[10] J.-P. Kahane; J. Peyrière Sur certaines martingales de Benoît Mandelbrot, Adv. Math., Volume 22 (1976), pp. 131-145

[11] D. Kuhlbusch On weighted branching processes in random environment, Stoch. Process. Appl., Volume 109 (2004) no. 1, pp. 113-144

[12] X. Liang Propriétés asymptotiques des martingales de Mandelbrot et des marches aléatoires branchantes, Université de Bretagne-Sud, France, 2010 (Thèse de doctorat)

[13] Q. Liu On generalized multiplicative cascades, Stoch. Process. Appl., Volume 86 (2000), pp. 263-286

[14] R.D. Mauldin; S.C. Williams Random recursive constructions: asymptotic geometric and topological properties, Trans. Am. Math. Soc., Volume 295 (1986) no. 1, pp. 325-346

[15] V.A. Topchii; V.A. Vatutin Maximum of the critical Galton–Watson processes and left continuous random walks, Theory Probab. Appl., Volume 42 (1997), pp. 17-27

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