Year: 2025
Type: article
Abstract: When it comes to random walk on the integers $${\mathbb {Z}}$$ 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 $${\mathbb {Z}}$$ Z started at 1 has a direct connection to the total progeny in branching processes. In the analysis of Maker–Breaker games on trees arising from these branching processes, however, the corresponding random walks have increments bounded from below by $$-2$$ - 2 instead of $$-1$$ - 1 . In this article, the probability of left-continuous random walk started at 0 to be negative at an even (resp. odd) time is derived and used to determine the probability of such nearly left-continuous random walk started at 0 to eventually become negative.
Author: Timo Vilkas
Institution: Lund University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
Funder: Lunds Universitet
OpenAlex ID: W4413681763