Convergence of complex martingales in supercritical multi-type general branching processes in $L^q$ for $1 < q \leq 2$

TL;DR

This paper extends the theory of martingale convergence in multi-type supercritical branching processes, establishing conditions for $L^q$ convergence of complex-valued martingales, which enhances understanding of process fluctuations.

Contribution

It constructs complex Nerman-type martingales for multi-type processes and provides new conditions for their $L^q$ convergence, advancing branching process theory.

Findings

Established $L^q$ convergence conditions for complex martingales

Extended Nerman's martingale framework to multi-type processes

Enhanced understanding of fluctuations in branching processes

Abstract

Nerman's martingale plays a central role in the law of large numbers for both, single- and multi-type, supercritical general branching processes. There are further, complex-valued Nerman-type martingales in the single-type process that figure in the finer fluctuations of these processes. We construct the analogous martingales for the process with finitely many types and give sufficient conditions for these martingales to converge in $L^q$ for $q \in (1,2]$.