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

Abstract

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