The critical Galton-Watson process vanishes

TL;DR

This paper provides a probabilistic proof of the classical result that a Galton-Watson process survives only if the average offspring exceeds one, offering an alternative to traditional analytic methods.

Contribution

It introduces a new probabilistic proof of the criticality condition for the Galton-Watson process, differing from classical generating function approaches.

Findings

Survival occurs only when the mean offspring exceeds one.

Provides a probabilistic proof in the spirit of Bezuidenhout and Grimmett.

Enhances understanding of Galton-Watson process criticality.

Abstract

The Galton-Watson process belongs to standard teaching in probability. The basic theorem says that survival is only possible when the fecondity exceeds 1. The classical proof is essentially analytic, using generating functions and convexity arguments. We give here a more probabilistic proof, in the spirit of Bezuidenhout and Grimmett.