Explicit stationary distribution of the $(L,1)$-reflecting random walk on the half line

TL;DR

This paper derives explicit criteria for recurrence and stationary distribution of a state-dependent reflecting random walk on the half line, revealing geometric tail behavior using an intrinsic branching structure.

Contribution

It provides the first explicit stationary distribution and recurrence criteria for the $(L,1)$-reflecting random walk, utilizing a novel branching structure approach.

Findings

Explicit recurrence criterion established

Stationary distribution explicitly derived

Geometric tail asymptotics proven

Abstract

In this paper, we consider the $(L,1)$ state-dependent reflecting random walk (RW) on the half line, which is a RW allowing jumps to the left at a maxial size $L$. For this model, we provide an explicit criterion for (positive) recurrence and an explicit expression for the stationary distribution.As an application, we prove the geometric tail asymptotic behavior of the stationary distribution under certain conditions. The main tool employed in the paper is the intrinsic branching structure within the $(L,1)$-random walk.