Stochastic Ordering of Infinite Geometric Galton-Watson Trees

TL;DR

This paper proves a stochastic ordering result for infinite Galton-Watson trees with geometric offspring distributions, showing a coupling exists that preserves subtree inclusion for different parameters.

Contribution

It establishes a coupling between infinite geometric Galton-Watson trees for parameters in [1/2, 1], demonstrating a stochastic ordering based on offspring probability.

Findings

Existence of a coupling for trees with different parameters

Subtree inclusion holds almost surely in the coupling

Stochastic ordering extends to infinite geometric Galton-Watson trees

Abstract

We consider Galton-Watson trees with Geom$(p)$ offspring distribution. We let $T_{\infty}(p)$ denote such a tree conditioned on being infinite. We prove that for any $1/2\leq p_1 <p_2 \leq 1$, there exists a coupling between $T_{\infty}(p_1)$ and $T_{\infty}(p_2)$ such that ${\mathbb P}(T_{\infty}(p_1) \subseteq T_{\infty}(p_2))=1.$