Left-continuous random walk on $\mathbb{Z}$ and the parity of its hitting times

TL;DR

This paper investigates the properties of left-continuous random walks on integers, deriving probabilities related to their negativity at even or odd times and connecting these to branching process progeny.

Contribution

It introduces new probability formulas for left-continuous random walks and links hitting times to branching process theory.

Findings

Derived probabilities for the walk being negative at even or odd times

Connected hitting time distributions to total progeny in branching processes

Provided conditions for the walk to eventually become negative

Abstract

When it comes to random walk on the integers $\mathbb{Z}$, the arguably first step of generalization beyond simple random walk is the class of one-sidedly continuous random walk, where the stepsize in only one direction is bounded by 1. Moreover, the time until state 0 is hit by left-continuous random walk on $\mathbb{Z}$ has a direct connection to the total progeny in branching processes. In this article, the probability of left-continuous random walk to be negative at an even (resp.\ odd) time is derived and used to determine the probability of nearly left-continuous random walk to eventually become negative.