Convergence of complex martingales in the branching random walk: the boundary

TL;DR

This paper extends the understanding of complex martingales in branching random walks by analyzing their behavior at the boundary of the convergence domain, revealing diverse convergence phenomena under weaker assumptions.

Contribution

It characterizes the boundary behavior of additive martingales in branching random walks, identifying regions of divergence, non-existence, and convergence, with improved moment conditions.

Findings

Martingales do not exist on some boundary parts.

Martingales converge to non-degenerate limits on other parts.

Weaker moment assumptions suffice for convergence results.

Abstract

Biggins [Uniform convergence of martingales in the branching random walk. {\em Ann. Probab.}, 20(1):137--151, 1992] proved local uniform convergence of additive martingales in $d$-dimensional supercritical branching random walks at complex parameters $\lambda$ from an open set $\Lambda \subseteq \mathbb{C}^d$. We investigate the martingales corresponding to parameters from the boundary $\partial \Lambda$ of $\Lambda$. The boundary can be decomposed into several parts. There may be a part of the boundary, on which the martingales do not exist, on other parts it exists, but diverges or vanishes in the limit. In the remaining part, there is convergence to a non-degenerate limit. The arguments that give this convergence also apply in $\Lambda$ and require weaker moment assumptions than the ones used by Biggins.