ASYMPTOTIC PROPERTIES OF A BRANCHING RANDOM WALK WITH A RANDOM ENVIRONMENT IN TIME

doi:10.1007/s10473-019-0513-y

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (5): 1345-1362.doi: 10.1007/s10473-019-0513-y

ASYMPTOTIC PROPERTIES OF A BRANCHING RANDOM WALK WITH A RANDOM ENVIRONMENT IN TIME

王月娇¹, 刘再明², 刘全升^3,4, 李应求^3,4

1. College of Mathematics and Computational Science, Hunan First Normal University, Changsha 410205, China;
2. School of Mathematics and Statistics, Central South University, Changsha 410083, China;
3. LMBA, UMR CNRS 6205, Université de Bretagne-Sud, F-56000 Vannes, France;
4. School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410004, China

收稿日期:2017-11-01 修回日期:2019-05-17 出版日期:2019-10-25 发布日期:2019-11-11
通讯作者: Quansheng LIU E-mail:quansheng.liu@univ-ubs.fr
作者简介:Yuejiao WANG,E-mail:wangyuejiaohujing@163.com;Zaiming LIU,E-mail:math_lzm@csu.edu.cn;Yingqiu LI,E-mail:liyq-2001@163.com
基金资助:
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).

ASYMPTOTIC PROPERTIES OF A BRANCHING RANDOM WALK WITH A RANDOM ENVIRONMENT IN TIME

Yuejiao WANG¹, Zaiming LIU², Quansheng LIU^3,4, Yingqiu LI^3,4

1. College of Mathematics and Computational Science, Hunan First Normal University, Changsha 410205, China;
2. School of Mathematics and Statistics, Central South University, Changsha 410083, China;
3. LMBA, UMR CNRS 6205, Université de Bretagne-Sud, F-56000 Vannes, France;
4. School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410004, China

Received:2017-11-01 Revised:2019-05-17 Online:2019-10-25 Published:2019-11-11
Contact: Quansheng LIU E-mail:quansheng.liu@univ-ubs.fr
Supported by:
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).

摘要/Abstract

摘要： We consider a branching random walk in an independent and identically distributed random environment ξ=(ξ_n) indexed by the time. Let W be the limit of the martingale W_n=∫e^-txZ_n(dx)/E_ξ ∫e^-txZ_n(dx), with Z_n denoting the counting measure of particles of generation n, and E_ξ 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.

关键词: branching random walk, random environment, quenched moments, weighted moments

Abstract: We consider a branching random walk in an independent and identically distributed random environment ξ=(ξ_n) indexed by the time. Let W be the limit of the martingale W_n=∫e^-txZ_n(dx)/E_ξ ∫e^-txZ_n(dx), with Z_n denoting the counting measure of particles of generation n, and E_ξ 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.

Key words: branching random walk, random environment, quenched moments, weighted moments

中图分类号:

60K37

王月娇, 刘再明, 刘全升, 李应求. ASYMPTOTIC PROPERTIES OF A BRANCHING RANDOM WALK WITH A RANDOM ENVIRONMENT IN TIME[J]. 数学物理学报(英文版), 2019, 39(5): 1345-1362.

Yuejiao WANG, Zaiming LIU, Quansheng LIU, Yingqiu LI. ASYMPTOTIC PROPERTIES OF A BRANCHING RANDOM WALK WITH A RANDOM ENVIRONMENT IN TIME[J]. Acta mathematica scientia,Series B, 2019, 39(5): 1345-1362.

参考文献

[1] Athreya K B, Karlin S. On branching processes in random environments. I:Extinction probabilities. Ann Math Statist, 1971, 42:1499-1520
[2] Athreya K B, Karlin S. On branching processes in random environments. II:Limit theorems. Ann Math Statist, 1971, 42:1843-1858
[3] Athreya K B, Ney P E. Branching Processes. Berlin:Springer, 1972
[4] Alsmeyer G, Rösler U. On the existence of φ-moments of the limit of a normalized supercritical GaltonWatson process. J Theor Probab, 2004, 17:905-928
[5] Alsmeyer G, Kuhlbusch D. Double martingale structure and existence of φ-moments for weighted branching processes. Münster J Math, 2010, 3:163-212
[6] Barral J. Generalized vector multiplicative cascades. Adv Appl Prob, 2001, 33:874-895
[7] Biggins J D. Martingale convergence in the branching random walk. J Appl Prob, 1977, 14(1):25-37
[8] Biggins J D. Uniform convergence of martingale in the branching random walk. Ann Prob, 1992, 20(1):137-151
[9] Biggins J D, Kyprianou A E. Measure change in multitype branching. Adv Appl Prob, 2004, 36(2):544-581
[10] Bingham N H, Doney R A. Asymptotic properties of supercritical branching processes. I:The GaltonWatson process. Adv Appl Prob, 1974, 6:711-731
[11] Bingham N H, Doney R A. Asymptotic properties of supercritical branching processes. II:Crump-Mode and Jirina process. Adv Appl Prob, 1975, 7:66-82
[12] Bingham K H, Goldie C M, Teugels J L. Regular Variation. Cambridge:Cambridge Univ Press, 1987
[13] Chen X, He H. On large deviation probabilities for empirical distribution of branching random walks:Schröder case and Böttcher case. 2017
[14] Chow Y S, Teicher H. Probability Theory:Independence, Interchangeability, Martingales. New York:Springer, 1995
[15] 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, 19:163-168
[16] Guivarc'h Y, Liu Q. Proprietes asympotiques des processus de branchement en environnement aleatoire. C R Acad Sci Paris, Ser I. 2001, 332:339-344
[17] Huang C. Limit Theorems and the Convergence Rate of Some Branching Processes and Branching Random Walk[D]. Universite de Bretagne-Sud (France), 2010
[18] Huang C, Liu Q. Convergence in Lp and its exponential rate for a branching process in a random environment. Electron J Prob, 2014, 104(19):1-22
[19] Hu Y, Shi Z. Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann Prob, 2009, 37:742-789
[20] Kuhlbusch D. On weighted branching process in random environment. Stoch Prob Appl, 2004, 109(1):113-144
[21] Li Y, Liu Q. Age-dependent Branching processes in random environments. Sci China Ser A, 2008, 51(10):1807-1830
[22] Liang X. Asymptotic Properties of the Mandelbrot's Martingale and the Branching Random Walks[D]. Universite de Bretagne-Sud (France), 2010
[23] Liang X, Liu Q. Weighted moments for the limit of a normalized supercritical Galton-Watson process. C R Acad Sci Paris, Ser I, 2013, 351:769-773
[24] Liang X, Liu Q. Weighted moments of the limit of a branching process in a random environment. Proc Steklov Inst Math, 2013, 282:127-145
[25] Liang X, Liu Q. Weighted moments for Mandelbrot's martingales. Electron Commun Probab, 2015, 20(85):1-12
[26] Liu Q. On generalized multiplicascades. Stoc Proc Appl, 2000, 86:263-286
[27] 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. 2007:702-219
[28] 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, 126:2634-2664
[29] 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
[30] Mandelbrot B. Multiplications aléatoires et distributions invaricantes par moyenne pondérée aléatoire. C R Acad Sci Pairs, 1974, 287:289-292, 355-358
[31] Shi Z. Branching Random Walks. Ecole d'Été de Probabilités de Saint-Flour XLII-2012. Lecture Notes in Mathematics, Vol 2151. Berlin:Springer, 2015
[32] 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

ASYMPTOTIC PROPERTIES OF A BRANCHING RANDOM WALK WITH A RANDOM ENVIRONMENT IN TIME

ASYMPTOTIC PROPERTIES OF A BRANCHING RANDOM WALK WITH A RANDOM ENVIRONMENT IN TIME

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 0

[1]	季丽娜, 郑祥祺. MOMENTS OF CONTINUOUS-STATE BRANCHING PROCESSES IN LÉVY RANDOM ENVIRONMENTS[J]. 数学物理学报(英文版), 2019, 39(3): 781-796.
[2]	高志强. A NOTE ON EXACT CONVERGENCE RATE IN THE LOCAL LIMIT THEOREM FOR A LATTICE BRANCHING RANDOM WALK[J]. 数学物理学报(英文版), 2018, 38(4): 1259-1268.
[3]	高志强, 刘全升, 汪和松. CENTRAL LIMIT THEOREMS FOR A BRANCHING RANDOM WALK WITH A RANDOM ENVIRONMENT IN TIME[J]. 数学物理学报(英文版), 2014, 34(2): 501-512.
[4]	洪文明, 孙鸿雁. RENEWAL THEOREM FOR (L, 1)-RANDOM WALK IN RANDOM ENVIRONMENT[J]. 数学物理学报(英文版), 2013, 33(6): 1736-1748.
[5]	高振龙, 胡晓予. LIMIT THEOREMS FOR A GALTON-WATSON PROCESS IN THE I.I.D. RANDOM ENVIRONMENT[J]. 数学物理学报(英文版), 2012, 32(3): 1193-1205.
[6]	毛明志, 李志民. ASYMPTOTIC BEHAVIOR FOR RANDOM WALK IN RANDOM \|ENVIRONMENT WITH HOLDING TIMES[J]. 数学物理学报(英文版), 2010, 30(5): 1696-1708.
[7]	王伟刚, 李燕, 胡迪鹤. EXTINCTION OF POPULATION-SIZE-DEPENDENT BRANCHING CHAINS IN RANDOM ENVIRONMENTS[J]. 数学物理学报(英文版), 2010, 30(4): 1065-1072.
[8]	毛明志, 韩东. LYAPOUNOV EXPONENTS AND LAW OF LARGE NUMBERS FOR RANDOM WALK IN RANDOM ENVIRONMENT WITH HOLDING TIMES[J]. 数学物理学报(英文版), 2009, 29(5): 1383-1394.
[9]	胡迪鹤; 胡晓予. ON MARKOV CHAINS IN SPACE-TIME RANDOM ENVIRONMENTS[J]. 数学物理学报(英文版), 2009, 29(1): 1-10.
[10]	胡迪鹤; 胡晓予. THE CONSTRUCTION OF DENUMERABLE q-PROCESSES IN RANDOM ENVIRONMENTS SATISFYING (F) OR (B)[J]. 数学物理学报(英文版), 2008, 28(4): 975-988.
[11]	胡迪鹤; 胡晓予. THE CONSTRUCTION OF DENUMERABLE q-PROCESSES IN RANDOM ENVIRONMENTS-THE EXISTENCE AND UNIQUENESS[J]. 数学物理学报(英文版), 2008, 28(2): 225-235.
[12]	胡迪鹤; 张书林. THE ERGODICITY FOR BI-IMMIGRATION BIRTH AND DEATH PROCESSES IN RANDOM ENVIRONMENT[J]. 数学物理学报(英文版), 2008, 28(1): 43-53.
[13]	胡迪鹤; 邱育峰. THE ANALYTICAL PROPERTIES FOR HOMOGENEOUS RANDOM TRANSITION FUNCTIONS[J]. 数学物理学报(英文版), 2007, 27(1): 180-192.
[14]	张晓敏; 胡迪鹤. THE DIMENSIONS OF THE RANGE OF RANDOM WALKS IN TIME-RANDOM ENVIRONMENTS[J]. 数学物理学报(英文版), 2006, 26(4): 615-628.
[15]	胡迪鹤. THE CONSTRUCTION OF MULTITYPE CANONICAL MARKOV BRANCHING CHAINS IN RANDOM ENVIRONMENTS[J]. 数学物理学报(英文版), 2006, 26(3): 431-442.