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 $(S_n^{\mathbf{h}}{)}_{n\ge 0}$ 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 $T^{\mathbf{h}}=\inf \{n\ge 1:S_n^{\mathbf{h}}=(0,\dots ,0)\}$, the first return time to the origin. For two $d$-dimensional probability vectors ${\mathbf{h}}^{\prime }$ and ${\mathbf{h}}^{\prime \prime }$ with the former majorizing the latter, we show that $T^{{\mathbf{h}}^{\prime }}$ is stochastically smaller than $T^{{\mathbf{h}}^{\prime \prime }}$. 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 $\mathbf{h}$.

26 pages