Extremum of a time-inhomogeneous branching random walk

doi:10.1007/s11464-021-0811-7

Front. Math. China

2021, Vol. 16

Issue (2) : 459-478 https://doi.org/10.1007/s11464-021-0811-7

RESEARCH ARTICLE

Extremum of a time-inhomogeneous branching random walk

Wanting HOU¹(

), Xiaoyue ZHANG², Wenming HONG³

¹. Department of Mathematics, Northeastern University, Shenyang 110004, China
². School of Statistics, Capital University of Economics and Business, Beijing 100070, China
³. School of Mathematical Sciences & Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing 100875, China

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Abstract

Consider a time-inhomogeneous branching random walk, generated by the point process L_n which composed by two independent parts: ‘branching’offspring X_n with the mean $1 + B (1 + n) − β$ for $β ∈ (0, 1)$ and ‘displacement’ $ξ n$ with a drift $A (1 + n) − 2 α$ for $α ∈ (0, 1 / 2)$ , where the ‘branching’ process is supercritical for B>0 but ‘asymptotically critical’ and the drift of the ‘displacement’ $ξ n$ is strictly positive or negative for $| A | 〉 0$ but ‘asymptotically’ goes to zero as time goes to infinity. We find that the limit behavior of the minimal (or maximal) position of the branching random walk is sensitive to the ‘asymptotical’ parameter $β$ and $α$ .

Keywords Branching random walk time-inhomogeneous branching process random walk

Corresponding Author(s): Wanting HOU

Issue Date: 01 June 2021

Cite this article:

Wanting HOU,Xiaoyue ZHANG,Wenming HONG. Extremum of a time-inhomogeneous branching random walk[J]. Front. Math. China, 2021, 16(2): 459-478.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-021-0811-7
https://academic.hep.com.cn/fmc/EN/Y2021/V16/I2/459

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