The Law of the Iterated Logarithm for the Single Point Range of Random Walk

Abstract

The number of the distinct points entered by a random walk once and only once in the first $n$ steps is called the single point range up to time $n$. We consider the random walk on the $d$ dimensional integer lattice. When $d\geq 4$, the author showed a limiting behavior of the variance of the single point range and established the central limit theorem. In this note, we proved the law of the iterated logarithm in the same case.