Beyond the Q-process: various ways of conditioning the multitype Galton-Watson process

TL;DR

This paper explores various conditioning methods for multitype Galton-Watson processes, extending the concept of the $Q$-process beyond indefinite survival to thresholds, states, and total progeny, revealing new limit behaviors.

Contribution

It introduces new conditioning frameworks for multitype Galton-Watson processes and analyzes their resulting $Q$-processes and stationary measures.

Findings

Conditioning on reaching a positive threshold yields a $Q$-process.

Stationary measure of the $Q$-process is a double limit.

Conditioning on infinite total progeny differs from the $Q$-process in non-critical regimes.

Abstract

Conditioning a multitype Galton-Watson process to stay alive into the indefinite future leads to what is known as its associated $Q$-process. We show that the same holds true if the process is conditioned to reach a positive threshold or a non-absorbing state. We also demonstrate that the stationary measure of the $Q$-process, obtained by construction as two successive limits (first by delaying the extinction in the original process and next by considering the long-time behavior of the obtained $Q$-process), is as a matter of fact a double limit. Finally, we prove that conditioning a multitype branching process on having an infinite total progeny leads to a process presenting the features of a $Q$-process. It does not however coincide with the original associated $Q$-process, except in the critical regime.