Joint Statistics of Random Walk on ${Z^1}$ and Accumulation of Visits

Abstract

We obtain the joint distribution $P_N(X,K\,|\,Z)$ of the location $X$ of a one-dimensional symmetric next neighbor random walk on the integer lattice, and the number of times the walk has visited a specified site $Z$. This distribution has a simple form in terms of the one variable distribution $p_{N'} (X')$, where $N'=N-K$ and $X'$ is a function of $X$, $K$, and $Z$. The marginal distributions of $X$ and $K$ are obtained, as well as their diffusion scaling limits.