Favorite sites of one-dimensional transient random walk

Abstract

In this paper, we study favorite sites of one-dimensional transient random walk on $\mathbb{Z}$ beyond the simple case. Denote by $\mathcal{K}(n)$ the set of favorite sites at time $n$. The goal of this paper is twofold. First, we study $\# \mathcal{K}(n)$, the cardinality of $\mathcal{K}(n)$, and show that, under some mild conditions, for any positive integer $k$, $\mathbf{P}(\#\mathcal{K}(n)=k, {\rm i.o.})=1$, and $\limsup_{n\to\infty}\#\mathcal{K}(n)/\log\log n=-1/\log \gamma$, where $\gamma$ is the probability that the random walk never returns to the starting point. Second, we study the escape rate of favorite sites, and give the limsup and liminf behaviors, where the liminf behavior is characterized via an integral test. To the best of our knowledge, this integral test is the first result of this nature for random walks (in non-random environments).