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 $${\mathbb {P}}(T_{\infty }(p_1) \subseteq T_{\infty }(p_2))=1$$ .