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

TL;DR

This paper derives the joint distribution of a one-dimensional symmetric random walk's position and visit count to a specific site, providing explicit formulas and diffusion limits.

Contribution

It presents a new explicit joint distribution formula for the walk's position and visit count, along with their marginal and scaling limit distributions.

Findings

Explicit joint distribution formula derived

Marginal distributions obtained

Diffusion scaling limits characterized

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 distribution of $X$ and $K$ are obtained, as well as their diffusion scaling limits.