依时变化的随机环境中的分枝随机游动的局部极限定理的二阶展开

数学物理学报 ›› 2022, Vol. 42 ›› Issue (1): 282-305.

依时变化的随机环境中的分枝随机游动的局部极限定理的二阶展开

高志强()

北京师范大学数学科学学院 & 教育部数学与复杂系统实验室北京 100875

收稿日期:2019-12-25 出版日期:2022-02-26 发布日期:2022-02-23
作者简介:高志强, E-mail: gaozq@bnu.edu.cn
基金资助:
国家自然科学基金(11971063)

A Second Order Correction of the Local Limit Theorem for a Branching Random Walk with a Random Environment in Time on ${\mathbb{Z}}^d$

Zhiqiang Gao()

Laboratory of Mathematics and Complex Systems(Ministry of Education) & School of Mathematical Sciences, Beijing Normal University, Beijing 100875

Received:2019-12-25 Online:2022-02-26 Published:2022-02-23
Supported by:
the NSFC(11971063)

摘要/Abstract

摘要：

考虑了${\mathbb{Z}}^d$中随机环境中的分枝随机游动, 其中分枝机制和粒子迁移的分布律均依时间变化.对任意给定点$z\in{\mathbb{Z}}^d$, 令$Z_n (z)$表示位于该点处的$n$代粒子的个数.给出了$Z_n (z)$的二阶渐近展开表达式.

关键词: 分枝随机游动, 局部极限定理, 渐近展开

Abstract:

Consider a branching random walk on ${\mathbb{Z}}^d$ with a random environment in time, where the branching offspring distribution and the migration law change as times goes by. Under the mild moment conditions, we derive the second order expansion for $Z_n(z)$, which counts the number of particles of generation $n$ at $z\in {\mathbb{Z}}^d$.

Key words: Branching random walk, Local limit theorem, Asymptotic expansions

中图分类号:

O211.6

高志强. 依时变化的随机环境中的分枝随机游动的局部极限定理的二阶展开[J]. 数学物理学报, 2022, 42(1): 282-305.

Zhiqiang Gao. A Second Order Correction of the Local Limit Theorem for a Branching Random Walk with a Random Environment in Time on ${\mathbb{Z}}^d$[J]. Acta mathematica scientia,Series A, 2022, 42(1): 282-305.

参考文献

1	Asmussen S, Hering H. Branching Processes. Boston: Birkh?user Boston, 1983
2	Révész P . Random Walks of Infinitely Many Particles. River Edge: World Scientific Publishing Co, 1994
3	Harris T E . The Theory of Branching Processes. Berlin: Springer-Verlag, 1963
4	Asmussen S , Kaplan N . Branching random walks. Ⅰ. Stochastic Process Appl, 1976, 4 (1): 1- 13
5	Gao Z Q , Liu Q , Wang H . Central limit theorems for a branching random walk with a random environment in time. Acta Math Sci, 2014, 34B (2): 501- 512
6	Chen X , He H . On large deviation probabilities for empirical distribution of supercritical branching random walks with unbounded displacements. Probab Theory Relat Fields, 2019, 175: 255- 307
7	Biggins J D . The central limit theorem for the supercritical branching random walk, and related results. Stochastic Process Appl, 1990, 34 (2): 255- 274
8	Huang C , Liang X , Liu Q . Branching random walks with random environments in time. Frontiers of Mathematics in China, 2014, 9 (4): 835- 842
9	Jagers P . Galton-Watson processes in varying environments. J Appl Probab, 1974, 11: 174- 178
10	Joffe A , Moncayo A R . Random variables, trees, and branching random walks. Adv Math, 1973, 10: 401- 416
11	Kabluchko Z . Distribution of levels in high-dimensional random landscapes. Ann Appl Probab, 2012, 22 (1): 337- 362
12	Kaplan N , Asmussen S . Branching random walks. Ⅱ. Stoch Process Appl, 1976, 4 (1): 15- 31
13	Louidor O , Perkins W . Large deviations for the empirical distribution in the branching random walk. Electron J Probab, 2015, 20 (18): 1- 19
14	Stam A J . On a conjecture by Harris. Z Wahrsch verw Geb, 1966, 5: 202- 206
15	Uchiyama K . Spatial growth of a branching process of particles living in $\mathbb{R}.d$. Ann Probab, 1982, 10 (4): 896- 918
16	Grübel R , Kabluchko Z . Edgeworth expansions for profiles of lattice branching random walks. Ann Inst H Poincaré Probab Statist, 2017, 53 (4): 2103- 2134
17	Chen X . Exact convergence rates for the distribution of particles in branching random walks. Ann Appl Probab, 2001, 11 (4): 1242- 1262
18	Gao Z Q . Exact convergence rate of the local limit theorem for branching random walks on the integer lattice. Stoch Process Appl, 2017, 127 (4): 1282- 1296
19	Gao Z Q . A second order asymptotic expansion in the local limit theorem for a simple branching random walk in $\mathbb{Z}^{d}$. Stoch Process Appl, 2018, 128 (12): 4000- 4017
20	Gao Z Q , Liu Q . Exact convergence rate in the central limit theorem for a branching random walk with a random environment in time. Stoch Process Appl, 2016, 126 (9): 2634- 2664
21	Gao Z Q , Liu Q . Second and third orders asymptotic expansions for the distribution of particles in a branching random walk with a random environment in time. Bernoulli, 2018, 24 (1): 772- 800
22	Liu Q . Branching random walks in random environment. Proceedings of the 4th International Congress of Chinese Mathematicians (ICCM 2007), 2007, 2: 702- 719
23	Wang X , Huang C . Convergence of martingale and moderate deviations for a branching random walk with a random environment in time. J Theoret Probab, 2017, 30 (3): 961- 995
24	Wang X , Huang C . Convergence of complex martingale for a branching random walk in a time random environment. Electron Commun Probab, 2019, 24 (41): 1- 14
25	Wang Y , Liu Z , Liu Q , Li Y . Asymptotic properties of a branching random walk with a random environment in time. Acta Math Sci, 2019, 39B (5): 1345- 1362
26	Baillon J B , Clément Ph , Greven A , den Hollander F . A variational approach to branching random walk in random environment. Ann Probab, 1993, 21 (1): 290- 317
27	Birkner M, Geiger J, Kersting G. Branching Processes in Random Environment-A View on Critical and Subcritical Cases//Deuschel J, Greven A. Interacting Stochastic Systems. Berlin: Springer, 2005: 269-291
28	Comets F , Popov S . On multidimensional branching random walks in random environment. Ann Probab, 2007, 35 (1): 68- 114
29	Comets F , Popov S . Shape and local growth for multidimensional branching random walks in random environment. ALEA Lat Am J Probab Math Stat, 2007, 3: 273- 299
30	Comets F , Yoshida N . Branching random walks in space-time random environment: survival probability, global and local growth rates. J Theoret Probab, 2011, 24 (3): 657- 687
31	Greven A , den Hollander F . Branching random walk in random environment: phase transitions for local and global growth rates. Probab Theory Related Fields, 1992, 91 (2): 195- 249
32	Hu Y , Yoshida N . Localization for branching random walks in random environment. Stochastic Process Appl, 2009, 119 (5): 1632- 1651
33	Nakashima M . Almost sure central limit theorem for branching random walks in random environment. Ann Appl Probab, 2011, 21 (1): 351- 373
34	Yoshida N . Central limit theorem for branching random walks in random environment. Ann Appl Probab, 2008, 18 (4): 1619- 1635
35	Biggins J D , Kyprianou A E . Measure change in multitype branching. Adv Appl Probab, 2004, 36 (2): 544- 581
36	Athreya K B , Karlin S . On branching processes with random environments. Ⅰ. Extinction probabilities. Ann Math Statist, 1971, 42: 1499- 1520
37	Athreya K B , Karlin S . On branching processes with random environments. Ⅱ. Limit theorems. Ann Math Statist, 1971, 42: 1843- 1858
38	Smith W L, Wilkinson W E. On branching processes in random environments. Ann Math Statist, 1969, series 40: 814-827
39	Tanny D . A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means. Stochastic Process Appl, 1988, 28 (1): 123- 139
40	Huang C , Liu Q . Convergence rates for a branching process in a random environment. Markov Process and Related Fields, 2014, 20 (2): 265- 286
41	Asmussen S . Convergence rates for branching processes. Ann Probab, 1976, 4 (1): 139- 146
42	Petrov V V . Limit Theorems of Probability Theory. Oxford: Clarendon Press, 1995
43	Durrett R . Probability: Theory and Examples. Belmont: Duxbury Press, 1996
44	Lawler G F , Limic V . Random Walk: A Modern Introduction. Cambridge: Cambridge University Press, 2010
45	Liang X , Liu Q . Weighted moments of the limit of a branching process in a random environment. Proceedings of the Steklov Institute of Mathematics, 2013, 282 (1): 127- 145
46	Biggins J D . Growth rates in the branching random walk. Z Wahrsch verw Geb, 1979, 48 (1): 17- 34

[1]	鲁世平, 任景莉, 葛渭高. 一类奇摄动泛函微分方程边值问题[J]. 数学物理学报, 2001, 21(zk): 591-597.
[2]	莫嘉琪. 奇摄动高阶非线性椭圆型方程非局部边值问题[J]. 数学物理学报, 1999, 19(3): 301-306.
[3]	林苏榕. 极限方程有奇性的一类六阶椭圆型方程混合边值问题解的渐近式[J]. 数学物理学报, 1998, 18(S1): 149-156.
[4]	张祥, 叶勤. 奇摄动向量椭圆型积分微分方程边值问题[J]. 数学物理学报, 1995, 15(3): 269-274.
[5]	林宗池. 极限方程有奇性的一类四阶椭圆型方程混合边值问题解的渐近式[J]. 数学物理学报, 1995, 15(2): 190-197.