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

Abstract

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

Key words： Branching random walk; random environment; moments; uniform convergence; complex martingale; L^{\alpha }-convergenc

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