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

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