Acta mathematica scientia, Series B >
CENTRAL LIMIT THEOREMS FOR A BRANCHING RANDOM WALK WITH A RANDOM ENVIRONMENT IN TIME
Received date: 2012-07-12
Revised date: 2013-11-28
Online published: 2014-03-20
Supported by
The project is partially supported by the National Natural Science Foundation of China (NSFC, 11101039, 11171044, 11271045), a cooperation program between NSFC and CNRS of France (11311130103), the Fundamental Research Funds for the Central Universities, and Hunan Provincial Natural Science Foundation of China (11JJ2001).
We consider a branching random walk with a random environment in time, in which the offspring distribution of a particle of generation n and the distribution of the displacements of its children depend on an environment indexed by the time n. The envi-ronment is supposed to be independent and identically distributed. For A （R, let Zn(A) be the number of particles of generation n located in A. We show central limit theorems for the counting measure Zn(·) with appropriate normalization.
GAO Zhi-Qiang , LIU Quan-Sheng , WANG He-Song . CENTRAL LIMIT THEOREMS FOR A BRANCHING RANDOM WALK WITH A RANDOM ENVIRONMENT IN TIME[J]. Acta mathematica scientia, Series B, 2014 , 34(2) : 501 -512 . DOI: 10.1016/S0252-9602(14)60023-0
[1] Asmussen S, Kaplan N. Branching random walks I. Stochastic Processes Appl, 1976, 4(1): 1–13
[2] Athreya K B, Karlin S. On branching processes with random environments I. Extinction probabilities. Ann Math Statist, 1971, 42: 1499–1520
[3] Athreya K B, Karlin S. On branching processes with random environments II. Limit theorems. Ann Math Statist, 1971, 42: 1843–1858
[4] Baillon J B, Cl´ement Ph, Greven A, et al. A variational approach to branching random walk in random environment. Ann Probab, 1993, 21(1): 290–317
[5] Biggins J D. Martingale convergence in the branching random walk. J Appl Probability, 1977,14(1): 25–37
[6] Biggins J D. The central limit theorem for the supercritical branching random walk and related results. Stochastic Process Appl, 1990, 34(2): 255–274
[7] Biggins J D, Kyprianou A E. Measure change in multitype branching. Adv in Appl Probab, 2004, 36(2): 544–581
[8] Durrett R. Probability: theory and examples. 2nd ed. Belmont, CA: Duxbury Press, 1996
[9] 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
[10] Harris T E. The theory of branching processes. Die Grundlehren der Mathematischen Wissenschaften, Bd. 119. Berlin: Springer-Verlag, 1963
[11] Kaplan N. A note on the branching random walk. J Appl Probab, 1982 19(2): 421–424
[12] Kaplan N, Asmussen S. Branching random walks II. Stochastic Processes Appl, 1976, 4(1): 15–31
[13] Klebaner C F. Branching random walk in varying environments. Adv in Appl Probab, 1982, 14(2): 359–367
[14] Liang X, Liu Q. Weighted moments of the limit of a branching process in a random environment. Proc Steklov Inst Math, 2013, 282(1): 127–145
[15] Liu Q. Branching random walks in random environment//Ji L, Liu K, Yang L, et al. Proceedings of the 4th International Congress of Chinese Mathematicians (ICCM 2007). 2007, 2: 702–219
[16] Petrov V V. Limit theorems of probability theory. Sequences of independent random variables. Oxford Studies in Probability. Vol. 4. New York: The Clarendon Press, 1995
[17] R´ev´esz P. Random walks of infinitely many particles. River Edge, NJ: World Scientific Publishing Co Inc, 1994
[18] Smith W L, Wilkinson W E. On branching processes in random environments. Ann Math Statist, 1969, 40: 814–827
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