Size-biased branching population measures and the multi-type $x\log x$ condition

TL;DR

This paper studies the $x ext{-} ext{log} ext{}x$ condition in multi-type branching processes, establishing when the intrinsic martingale converges and exploring the condition's necessity for non-degenerate limits.

Contribution

It introduces a size-biased population measure for multi-type branching processes and proves sufficiency and investigates necessity of the $x ext{-} ext{log} ext{}x$ condition for martingale convergence.

Findings

Proves sufficiency of the $x ext{-} ext{log} ext{}x$ condition for non-degenerate limits.

Constructs a size-biased measure related to the population process.

Analyzes conditions under which the $x ext{-} ext{log} ext{}x$ condition is necessary.

Abstract

We investigate the $x\log x$ condition for a general (Crump--Mode--Jagers) multi-type branching process with a general type space by constructing a size-biased population measure that relates to the ordinary population measure via an intrinsic martingale $W_t$. Sufficiency of the $x\log x$ condition for a non-degenerate limit of $W_t$ is proved and conditions for necessity are investigated.