The Speed of a Biased Walk on a Galton-Watson Tree without Leaves is Monotonic with Respect to Progeny Distributions for High Values of Bias

Consider biased random walks on two Galton-Watson trees without leaves having progeny distributions $P_1$ and $P_2$ (GW$(P_1)$ and GW$(P_2)$) where $P_1$ and $P_2$ are supported on positive integers and $P_1$ dominates $P_2$ stochastically. We prove that the speed of the walk on GW$(P_1)$ is bigger than the same on GW$(P_2)$ when the bias is larger than a threshold depending on $P_1$ and $P_2$. This partially answers a question raised by Ben Arous, Fribergh and Sidoravicius.