Acta mathematica scientia,Series A ›› 2022, Vol. 42 ›› Issue (1): 282-305.
Previous Articles Next Articles
Received:
2019-12-25
Online:
2022-02-26
Published:
2022-02-23
Supported by:
CLC Number:
Zhiqiang Gao. A Second Order Correction of the Local Limit Theorem for a Branching Random Walk with a Random Environment in Time on
Add to citation manager EndNote|Reference Manager|ProCite|BibTeX|RefWorks
1 | Asmussen S, Hering H. Branching Processes. Boston: Birkhäuser 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
doi: 10.1016/0304-4149(76)90022-3 |
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
doi: 10.1007/s00440-018-0891-4 |
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
doi: 10.1016/0304-4149(90)90018-N |
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
doi: 10.1007/s11464-014-0407-1 |
9 |
Jagers P . Galton-Watson processes in varying environments. J Appl Probab, 1974, 11: 174- 178
doi: 10.2307/3212594 |
10 |
Joffe A , Moncayo A R . Random variables, trees, and branching random walks. Adv Math, 1973, 10: 401- 416
doi: 10.1016/0001-8708(73)90123-0 |
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
doi: 10.1016/0304-4149(76)90023-5 |
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
doi: 10.1007/BF00533055 |
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
doi: 10.1016/j.spa.2016.07.015 |
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
doi: 10.1016/j.spa.2018.01.005 |
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
doi: 10.1016/j.spa.2016.02.013 |
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
doi: 10.1007/s10959-016-0668-6 |
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
doi: 10.1007/s10959-009-0267-x |
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
doi: 10.1007/BF01291424 |
32 |
Hu Y , Yoshida N . Localization for branching random walks in random environment. Stochastic Process Appl, 2009, 119 (5): 1632- 1651
doi: 10.1016/j.spa.2008.08.005 |
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
doi: 10.1017/S0001867800013604 |
36 |
Athreya K B , Karlin S . On branching processes with random environments. Ⅰ. Extinction probabilities. Ann Math Statist, 1971, 42: 1499- 1520
doi: 10.1214/aoms/1177693150 |
37 |
Athreya K B , Karlin S . On branching processes with random environments. Ⅱ. Limit theorems. Ann Math Statist, 1971, 42: 1843- 1858
doi: 10.1214/aoms/1177693051 |
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
doi: 10.1016/0304-4149(88)90070-1 |
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
doi: 10.1134/S0081543813060126 |
46 |
Biggins J D . Growth rates in the branching random walk. Z Wahrsch verw Geb, 1979, 48 (1): 17- 34
doi: 10.1007/BF00534879 |
[1] | Zhang Chuanzhou, Li Tiantian, Jiao Fan. Atomic Aecompositions of $B$-Valued Weak Orlicz $\alpha$-Quasi-Martingale Spaces [J]. Acta mathematica scientia,Series A, 2022, 42(1): 9-17. |
[2] | Jiang Li,Guijie Lan,Shuwen Zhang,Chunjin Wei. Dynamics Analysis of a Stochastic Glucose-Insulin Model [J]. Acta mathematica scientia,Series A, 2021, 41(6): 1937-1949. |
[3] | Liya Liu,Daqing Jiang. Global Dynamics of a Stochastic Chemostat Model with General Response Function and Wall Growth [J]. Acta mathematica scientia,Series A, 2021, 41(6): 1912-1924. |
[4] | Zhonghua Zhang,Qian Zhang. Qualitative Analysis of a Stochastic SIVS Epidemic Model with Nonlinear Perturbations Under Regime Switching [J]. Acta mathematica scientia,Series A, 2021, 41(4): 1218-1234. |
[5] | Yi Ding,Jingjun Guo. Pricing Asian Options Under Time-Changed Mixed Fractional Brownian Motion with Transactions Costs [J]. Acta mathematica scientia,Series A, 2021, 41(4): 1135-1146. |
[6] | Yunguo Lin. The Analysis of Evolution Process in a Time-Inhomogeneous Two-State Quantum Walk [J]. Acta mathematica scientia,Series A, 2021, 41(4): 1097-1110. |
[7] | Xiaofeng Yang,Hua Dong,Hongshuai Dai. Parisian Ruin for Spectrally Negative Lévy Processes Under a Hybrid Observation Scheme [J]. Acta mathematica scientia,Series A, 2021, 41(2): 548-561. |
[8] | Xuyan Xiang,Haiqin Fu,Jieming Zhou,Yingchun Deng,Xiangqun Yang. Statistical Identification of Reversible Markov Chain on Cyclic Graph [J]. Acta mathematica scientia,Series A, 2020, 40(6): 1682-1698. |
[9] | Linjuan Pu,Xiaozhong Yang,Shuzhen Sun. Numerical Analysis of a Class of Fractional Langevin Equation by Predictor-Corrector Method [J]. Acta mathematica scientia,Series A, 2020, 40(4): 1018-1028. |
[10] | Liheng Sang,Zhenlong Chen,Xiaozhen Hao. Smoothness for the Renormalized Self-Intersection Local Time of Bifractional Brownian Motion [J]. Acta mathematica scientia,Series A, 2020, 40(3): 796-810. |
[11] | Yingchun Deng,Man Li,Ya Huang,Jieming Zhou. On the Analysis of Ruin-Related Quantities in the Nonhomogeneous Compound Poisson Risk Model [J]. Acta mathematica scientia,Series A, 2020, 40(2): 501-514. |
[12] | Zhongwei Cao,Xiangdan Wen,Wei Feng,Li Zu. Dynamics of a Nonautonomous SIRI Epidemic Model with Random Perturbations [J]. Acta mathematica scientia,Series A, 2020, 40(1): 221-233. |
[13] | Qikang Ran. SDE Driven by Fractional Brown Motion and Their Coefficients are Locally Linear Growth [J]. Acta mathematica scientia,Series A, 2020, 40(1): 200-211. |
[14] | Chen Fei,Weiyin Fei. Consistency of Least Squares Estimation to the Parameter for Stochastic Differential Equations Under Distribution Uncertainty [J]. Acta mathematica scientia,Series A, 2019, 39(6): 1499-1513. |
[15] | Zhaoqiang Yang. Pricing European Lookback Option in a Special Kind of Mixed Jump-Diffusion Black-Scholes Model [J]. Acta mathematica scientia,Series A, 2019, 39(6): 1514-1531. |
|