Number of Distinct Sites Visited by a Random Walk with Internal States

TL;DR

This paper generalizes classical results on the number of distinct sites visited by a simple symmetric random walk to random walks with internal states, proving laws of large numbers and estimating error terms of local limit theorems.

Contribution

It extends asymptotic results and laws of large numbers from simple symmetric random walks to those with internal states, including error term estimates.

Findings

Asymptotic formulas for expected value and variance for walks with internal states

Proofs of weak and strong laws of large numbers in this context

Error term estimates for the local limit theorem

Abstract

In the classical paper of Dvoretzky-Erd\H{o}s, asymptotics for the expected value and the variance of the number of distinct sites visited by a Simple Symmetric Random Walk were calculated. Here, these results are generalized for Random Walks with Internal States. Moreover, both weak and strong laws of large numbers are proved. As a tool for these results, the error term of the local limit theorem in of Kr\'amli and Sz\'asz is also estimated.