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

• Published in: Frontiers of mathematics in China Vol. 16; no. 1; pp. 187 - 209
• Main Authors: WANG, Xin, LIANG, Xingang, HUANG, Chunmao
• Format: Journal Article
• Language: English
• Published: Beijing Higher Education Press 01.02.2021; Springer Nature B.V
• Subjects: Branching random walk; complex martingale; Convergence; L α -convergenc; Martingales; Mathematics; Mathematics and Statistics; moments; random environment; Random walk; Research Article; uniform convergence; moments; convergence; Branching random walk; random environment; uniform convergence; 60J80; 60K37; complex martingale

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...