Left-continuous random walk on ℤ and the parity of its hitting times

2024/07/09 by Timo Vilkas, Vilkas, Timo
Mathematics · #60G50 #60J10 #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2407.06903

openalex publication_date 2024/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

When it comes to random walk on the integers 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 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.

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