Convergence of complex martingale for a branching random walk in an independent and identically distributed environment

doi:10.1007/s11464-021-0882-0

Front. Math. China

2021, Vol. 16

Issue (1) : 187-209 https://doi.org/10.1007/s11464-021-0882-0

RESEARCH ARTICLE

Convergence of complex martingale for a branching random walk in an independent and identically distributed environment

Xin WANG¹, Xingang LIANG², Chunmao HUANG³(

)

¹. Department of General Education, Wenzhou Business College, Wenzhou 325035, China
². School of Mathematics and Statistics, Beijing Technology and Business University, Beijing 100048, China
³. Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, China

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Abstract

We consider an $ℝ d$ -valued discrete time branching random walk in an independent and identically distributed environment indexed by time $n ∈ ℕ$ . Let $W n (z) (z ∈ ℂ d)$ be the natural complex martingale of the process. We show necessary and sufficient conditions for the $L α$ -convergence of $W n (z)$ for $α$ >1, as well as its uniform convergence region.

Keywords Branching random walk random environment moments uniform convergence complex martingale Lα-convergenc')" href="#">

L α

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Corresponding Author(s): Chunmao HUANG

Issue Date: 26 March 2021

Cite this article:

Xin WANG,Xingang LIANG,Chunmao HUANG. Convergence of complex martingale for a branching random walk in an independent and identically distributed environment[J]. Front. Math. China, 2021, 16(1): 187-209.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-021-0882-0
https://academic.hep.com.cn/fmc/EN/Y2021/V16/I1/187

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