Asymptotic behavior of random walks on a half-line with a jump at the origin

TL;DR

This paper analyzes the long-term behavior of a discrete-time random walk on non-negative integers with jumps at zero, deriving an asymptotic formula for the expected position based on jump probabilities and initial state.

Contribution

It introduces a spectral analysis approach to derive the asymptotic behavior of the random walk with jumps at the origin, highlighting the role of eigenvalues and resonances.

Findings

Asymptotic formula for expected position at large times

Spectral analysis of transition operator reveals eigenvalues and resonances

Dependence of asymptotics on jump probabilities and initial position

Abstract

We study a discrete-time random walk on the non-negative integers, such that when 0 is reached a jump occurs to an arbitrary location, with given probabilities. We obtain an asymptotic formula for the expected position at large times, in dependence on the jump probabilities and on the starting position. Our proof of this result displays the relevance of the spectral analysis of the transition operator associated to the stochastic process, both of its eigenvalues and of its resonances.