Stochastic Ordering of 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 $1/2\le p_1<p_2\le 1$, there exists a coupling between $T_{\infty }(p_1)$ and $T_{\infty }(p_2)$ such that $\mathbb{P}(T_{\infty }(p_1)\subseteq T_{\infty }(p_2))=1.$

14 pages