Stochastic dominance of first return times for nearest-neighbor random walks on $\mathbb{Z}^d$

Abstract

For a $d$-dimensional probability vector $\mathbf{h}=(h_1,\dots, h_d)$, let $(S^{\mathbf{h}}_n)_{n\geq 0}$ be a nearest-neighbor random walk on $\mathbb{Z}^d$ such that at each step, it moves to one of the two nearest neighbors in the $i$-th dimension with probability $\frac{1}{2} h_i$ ($i=1,\dots, d$). Let $T^{\mathbf{h}}=\inf\{n\geq 1: S^{\mathbf{h}}_n=(0,\dots,0)\}$, the first return time to the origin. For two $d$-dimensional probability vectors $\mathbf{h}'$ and $\mathbf{h}''$ with the former majorizing the latter, we show that $T^{\mathbf{h}'}$ is stochastically smaller than $T^{\mathbf{h}''}$. In particular, the first return time for the $d$-dimensional simple random walk is stochastically larger than $T^{\mathbf{h}}$ for all $d$-dimensional probability vectors $\mathbf{h}$.