Yaglom's limit for critical Galton-Watson processes in varying environment: A probabilistic approach

TL;DR

This paper proves a probabilistic Yaglom limit for critical Galton-Watson processes in varying environments, showing that conditioned on survival, the process converges to an exponential distribution, using a two-spine decomposition approach.

Contribution

It introduces a probabilistic proof of the Yaglom limit for these processes, extending prior analytic results with a novel two-spine decomposition technique.

Findings

Conditioned process converges to exponential distribution

Probabilistic approach complements analytic methods

Applicable to critical Galton-Watson processes in varying environments

Abstract

A Galton-Watson process in varying environment is a discrete time branching process where the offspring distributions vary among generations. Based on a two-spine decomposition technique, we provide a probabilistic argument of a Yaglom-type limit for this family processes. The result states that, in the critical case, a suitable normalisation of the process conditioned on non-extinction converges in distribution to an exponential random variable. Recently, this result has been established by Kersting [{\it J. Appl. Probab.} {\bf57}(1), 196--220, 2020] using analytic techniques.