The mean number of sites visited by a random walk pinned at a distant point

Abstract

This paper concerns the number $Z_n$ of sites visited up to time $n$ by a random walk $S_n$ having zero mean and moving on the two dimensional square lattice ${\bf Z}^2$. Asymptotic evaluation of the conditional expectation of $Z_n$ for large $n$ given that $S_n=x$ is carried out under some exponential moment condition. It gives an explicit form of the leading term valid uniformly in $(x, n)$, $|x|< cn$ for each $c>0$.