Abstract
We consider a branching random walk in an independent and identically distributed random environment ξ = (ξn) indexed by the time. Let W be the limit of the martingale \(W_n=\int\;e^{-tx}Z_n(\text{d}x)/\mathbb{E}_\xi\int\;e^{-tx}Z_n(\text{d}x)\), with Zn denoting the counting measure of particles of generation n, and \(\mathbb{E}_\xi\) the conditional expectation given the environment ξ. We find necessary and sufficient conditions for the existence of quenched moments and weighted moments of W, when W is non-degenerate.
Similar content being viewed by others
References
Athreya K B, Karlin S. On branching processes in random environments. I: Extinction probabilities. Ann Math Statist, 1971, 42: 1499–152
Athreya K B, Karlin S. On branching processes in random environments. II: Limit theorems. Ann Math Statist, 1971, 42: 1843–185
Athreya K B, Ney P E. Branching Processes. Berlin: Springer, 1972
Alsmeyer G, Rösler U. On the existence of ø-moments of the limit of a normalized supercritical Galton- Watson process. J Theor Probab, 2004, 17: 905–92
Alsmeyer G, Kuhlbusch D. Double martingale structure and existence of ø-moments for weighted branching processes. M¨unster J Math, 2010, 3: 163–21
Barral J. Generalized vector multiplicative cascades. Adv Appl Prob, 2001, 33: 874–89
Biggins J D. Martingale convergence in the branching random walk. J Appl Prob, 1977, 14(1): 25–37
Biggins J D. Uniform convergence of martingale in the branching random walk. Ann Prob, 1992, 20(1): 137–151
Biggins J D, Kyprianou A E. Measure change in multitype branching. Adv Appl Prob, 2004, 36(2): 544–581
Bingham N H, Doney R A. Asymptotic properties of supercritical branching processes. I: The Galton-Watson process. Adv Appl Prob, 1974, 6: 711–73
Bingham N H, Doney R A. Asymptotic properties of supercritical branching processes. II: Crump-Mode and Jirina process. Adv Appl Prob, 1975, 7: 66–8
Bingham K H, Goldie C M, Teugels J L. Regular Variation. Cambridge: Cambridge Univ Press, 1987
Chen X, He H. On large deviation probabilities for empirical distribution of branching random walks: Schröder case and Böttcher case. 2017
Chow Y S, Teicher H. Probability Theory: Independence, Interchangeability, Martingales. New York: Springer, 1995
Grintsevichyus A K. On the continuity of the distribution of a sum of dependent variables connected with independent walks on lines. Theory Prob Appl, 1974, 19: 163–16
Guivarc’h Y, Liu Q. Proprietes asympotiques des processus de branchement en environnement aleatoire. C R Acad Sci Paris, Ser I. 2001, 332: 339–34
Huang C. Limit Theorems and the Convergence Rate of Some Branching Processes and Branching Random Walk [D]. Universite de Bretagne-Sud (France), 2010
Huang C, Liu Q. Convergence in L p and its exponential rate for a branching process in a random environment. Electron J Prob, 2014, 104(19): 1–22
Hu Y, Shi Z. Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann Prob, 2009, 37: 742–78
Kuhlbusch D. On weighted branching process in random environment. Stoch Prob Appl, 2004, 109(1): 113–144
Li Y, Liu Q. Age-dependent Branching processes in random environments. Sci China Ser A, 2008, 51(10): 1807–1830
Liang X. Asymptotic Properties of the Mandelbrot’s Martingale and the Branching Random Walks [D]. Universite de Bretagne-Sud (France), 2010
Liang X, Liu Q. Weighted moments for the limit of a normalized supercritical Galton-Watson process. C R Acad Sci Paris, Ser I, 2013, 351: 769–77
Liang X, Liu Q. Weighted moments of the limit of a branching process in a random environment. Proc Steklov Inst Math, 2013, 282: 127–14
Liang X, Liu Q. Weighted moments for Mandelbrot’s martingales. Electron Commun Probab, 2015, 20(85): 1–12
Liu Q. On generalized multiplicascades. Stoc Proc Appl, 2000, 86: 263–28
Liu Q. Branching random walks in random environment // Ji L, Liu K, Yang L, Yau S T, eds. Proceedings of the 4th International Congress of Chinese Mathematicians (ICCM 2007), Vol II. 2007: 702–21
Gao Z, Liu Q. Exact convergence rates in central limit theorems for a branching random walk with a random environment in time. Stoch Proc Appl, 2016, 126: 2634–266
Gao Z, Liu Q, Wang H. Central limit theorems for a branching random walk with a random environment in time. Acta Mathematica Scientia, 2014, 34B(2): 501–512
Mandelbrot B. Multiplications aléatoires et distributions invaricantes par moyenne pondérée aléatoire. C R Acad Sci Pairs, 1974, 287: 289–292: 355–358
Shi Z. Branching Random Walks. ´Ecole d’Été de Probabilités de Saint-Flour XLII - 2012. Lecture Notes in Mathematics, Vol 2151. Berlin: Springer, 2015
Wang Y, Li Y, Liu Q, Liu Z. Quenched weighted moments of a supercritical branching process in a random environment. Published in Asian Journal of Mathematics, 2019
Author information
Authors and Affiliations
Corresponding author
Additional information
The work has benefited from the support of the French government Investissements d’Avenir program ANR-11-LABX-0020-01. It has been partially supported by the National Natural Science Foundation of China (11571052, 11401590, 11731012 and 11671404), and by Hunan Natural Science Foundation (2017JJ2271).
Rights and permissions
About this article
Cite this article
Wang, Y., Liu, Z., Liu, Q. et al. Asymptotic Properties of a Branching Random Walk with a Random Environment in Time. Acta Math Sci 39, 1345–1362 (2019). https://doi.org/10.1007/s10473-019-0513-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-019-0513-y