A NOTE ON EXACT CONVERGENCE RATE IN THE LOCAL LIMIT THEOREM FOR A LATTICE BRANCHING RANDOM WALK

doi:10.1016/S0252-9602(18)30812-9

Acta mathematica scientia,Series B ›› 2018, Vol. 38 ›› Issue (4): 1259-1268.doi: 10.1016/S0252-9602(18)30812-9

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A NOTE ON EXACT CONVERGENCE RATE IN THE LOCAL LIMIT THEOREM FOR A LATTICE BRANCHING RANDOM WALK

Zhiqiang GAO

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

Received:2017-01-09 Revised:2017-12-15 Online:2018-08-25 Published:2018-08-25
Supported by:
The author was partially supported by NSFC (11101039).

Abstract

Abstract:

Consider a branching random walk, where the underlying branching mechanism is governed by a Galton-Watson process and the moving law of particles by a discrete random variable on the integer lattice Z. Denote by Z_n(z) the number of particles in the n-th generation in the model for each z ∈ Z. We derive the exact convergence rate in the local limit theorem for Z_n(z) assuming a condition like "EN(log N)^1+λ < ∞" for the offspring distribution and a finite moment condition on the motion law. This complements the known results for the strongly non-lattice branching random walk on the real line and for the simple symmetric branching random walk on the integer lattice.

Key words: lattice branching random walks, local limit theorem, exact convergence rate

Zhiqiang GAO. A NOTE ON EXACT CONVERGENCE RATE IN THE LOCAL LIMIT THEOREM FOR A LATTICE BRANCHING RANDOM WALK[J].Acta mathematica scientia,Series B, 2018, 38(4): 1259-1268.

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References 25

[1]	Asmussen S. Convergence rates for branching processes. Ann Probab, 1976, 4(1):139-146
[2]	Asmussen S, Hering H. Branching Processes. Boston, MA:Birkhäuser Boston, Inc, 1983
[3]	Asmussen S, Kaplan N. Branching random walks I. Stoch Process Appl, 1976, 4(1):1-13
[4]	Athreya K B, Ney P E. Branching Processes. New York:Springer-Verlag, 1972
[5]	Baum L E, Katz M. Convergence rates in the law of large numbers. Trans Amer Math Soc, 1965, 120:108-123
[6]	Biggins J D. Growth rates in the branching random walk. Z Wahrsch Verw Geb, 1979, 48(1):17-34
[7]	Biggins J D. The central limit theorem for the supercritical branching random walk, and related results. Stoch Process Appl, 1990, 34(2):255-274
[8]	Bingham N H, Doney R A. Asymptotic properties of supercritical branching processes, I, the Galton-Watson process. Adv Appl Probab, 1974, 6:711-731
[9]	Chen X. Exact convergence rates for the distribution of particles in branching random walks. Ann Appl Probab, 2001, 11(4):1242-1262
[10]	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
[11]	Gao Z-Q, Liu Q. Second and third orders asymptotic expansions for the distributionof particles in a branching random walk with a random environment in time. Bernoulli, 2018, 24(1):772-800
[12]	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
[13]	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
[14]	Grübel R, Kabluchko Z. Edgeworth expansions for profiles of lattice branching random walks. Ann Inst H Poincaré Probab Statist, 2017, 53(4):2103-2134
[15]	Gut A. Stopped Random Walks:Limit Theorems and Applications. 2nd ed. New York:Springer, 2009
[16]	Harris T E. The Theory of Branching Processes. Berlin:Springer-Verlag, 1963
[17]	Huang C, Liang X, Liu Q. Branching random walks with random environments in time. Frontiers of Mathematics in China, 2014, 9(4):835-842
[18]	Kaplan N, Asmussen S. Branching random walks Ⅱ. Stoch Process Appl, 1976, 4(1):15-31
[19]	Révész P. Random Walks of Infinitely Many Particles. River Edge, NJ:World Scientific Publishing Co Inc, 1994
[20]	Yoshida N. Central limit theorem for branching random walks in random environment. Ann Appl Probab, 2008, 18(4):1619-1635
[21]	Nakashima M. Almost sure central limit theorem for branching random walks in random environment. Ann Appl Probab, 2011, 21(1):351-373
[22]	Petrov V V. Sums of Independent Random Variables. New York, Heidelberg:Springer-Verlag, 1975
[23]	Shi Z. Branching Random Walks. Lecture Notes in Mathematics, Vol 2151. Cham:Springer, 2015
[24]	Stam A J. On a conjecture by Harris. Z Wahrsch Verw Geb, 1966, 5:202-206
[25]	Zeitouni O. Branching random walks and Gaussian fields. Lecture Notes, 2012

A NOTE ON EXACT CONVERGENCE RATE IN THE LOCAL LIMIT THEOREM FOR A LATTICE BRANCHING RANDOM WALK

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