Snakes and perturbed random walks

TL;DR

This paper investigates properties of perturbed random walks, including excited walks, establishing laws of large numbers, central limit theorems, and recurrence criteria for processes with finite and infinite memory.

Contribution

It introduces new results on the range, recurrence, and limit behaviors of perturbed random walks, extending previous models like excited random walks.

Findings

Proved the Strong Law of Large Numbers for the process.

Established the Central Limit Theorem for the perturbed walk.

Derived criteria for recurrence with finite memory.

Abstract

In this paper we study some properties of random walks perturbed at extrema, which are generalizations of the walks considered e.g., in Davis (1999). This process can also be viewed as a version of {\em excited random walk}, studied recently by many authors. We obtain a few properties related to the range of the process with infinite memory. We also prove the Strong law, Central Limit Theorem, and the criterion for the recurrence of the perturbed walk with finite memory.