Laws of the iterated logarithm for the range of once-reinforced random walk on $\mathbb{Z}$

Abstract

In this paper, we study once-reinforced random walk on $\mathbb{Z}$, and prove sharp Chung-type and Hartman-Wintner laws of the iterated logarithm for the range. Let $R_n$ denote the range at time $n$ and let $c>0$ be the reinforcement parameter. Then, almost surely, \begin{equation*} \liminf_{n\to\infty}\frac{R_n}{\sqrt{n/\log\log n}}=\fracπ{\sqrt{2}}, \qquad \limsup_{n\to\infty}\frac{cR_n}{\sqrt{2n\log\log n}}=1. \end{equation*} The proofs are based on the independent increment decomposition of the inverse range and sharp generating function estimates.