Light-tailed behavior of stationary distribution for state-dependent random walks on a strip

TL;DR

This paper analyzes a state-dependent reflecting random walk on a half-strip, providing explicit criteria for recurrence, stationary distribution, and demonstrating light-tailed behavior under certain conditions using a novel branching structure approach.

Contribution

It introduces a new method based on intrinsic branching structures to analyze the stationary distribution of state-dependent random walks on a strip, differing from traditional matrix-analytic techniques.

Findings

Explicit criteria for recurrence are established.

An explicit formula for the stationary distribution is derived.

Light-tailed behavior of the stationary distribution is proved.

Abstract

In this paper, we consider the state-dependent reflecting random walk on a half-strip. We provide explicit criteria for (positive) recurrence, and an explicit expression for the stationary distribution. As a consequence, the light-tailed behavior of the stationary distribution is proved under appropriate conditions. The key idea of the method employed here is the decomposition of the trajectory of the random walk and the main tool is the intrinsic branching structure buried in the random walk on a strip, which is different from the matrix-analytic method.