Persistent random walks

TL;DR

This paper analyzes persistent random walks where the probability of continuing in the same direction depends on the time spent moving, exploring their long-term behavior and recurrence conditions without assuming stationary probabilities.

Contribution

It provides new insights into the asymptotic properties and recurrence criteria of persistent random walks with non-stationary transition probabilities.

Findings

Conditions for recurrence and transience are characterized.

Examples show recurrence even in non-symmetric cases.

Abstract

We consider a walker that at each step keeps the same direction with a probabilitythat depends on the time already spent in the direction the walker is currently moving. In this paper, we study some asymptotic properties of this persistent random walk and give the conditions of recurrence or transience in terms of "transition" probabilities to keep on the same direction or to change, without assuming that the latter admits any stationary probability. Examples are exhibited when this process is recurrent even if the random walk is not symmetric.