Stochastic Ordering of Infinite Binomial Galton-Watson Trees

Abstract

We consider Galton-Watson trees with $\mathrm{Bin}(d,p)$ offspring distribution. We let $T_{\infty }(p)$ denote such a tree conditioned on being infinite. For $d=2,3$ and any $1/d\le p_1<p_2\le 1$, we show that there exists a coupling between $T_{\infty }(p_1)$ and $T_{\infty }(p_2)$ such that $\mathbb{P}(T_{\infty }(p_1)\subseteq T_{\infty }(p_2))=1.$

19 pages