Left-Continuous Random Walk on Z and the Parity of Its Hitting Times
Vilkas, Timo LU
(2025)
In Journal of Theoretical Probability
38(4).
- 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 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 instead of -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... (More)
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 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 instead of -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.
(Less)
- author
- Vilkas, Timo
LU
- organization
- publishing date
- 2025
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Branching process, Hitting time, Left-continuous random walk, Parity, Positive drift, Separable distribution, Skip-free
- in
- Journal of Theoretical Probability
- volume
- 38
- issue
- 4
- article number
- 68
- publisher
- Springer
- external identifiers
-
- scopus:105014149768
- ISSN
- 0894-9840
- DOI
- 10.1007/s10959-025-01440-x
- language
- English
- LU publication?
- yes
- id
- bd603b64-e4c3-4a02-8aae-9ac36038eb32
- date added to LUP
- 2025-10-03 09:55:50
- date last changed
- 2025-10-03 09:56:57
@article{bd603b64-e4c3-4a02-8aae-9ac36038eb32,
abstract = {{<p>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 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 instead of -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.</p>}},
author = {{Vilkas, Timo}},
issn = {{0894-9840}},
keywords = {{Branching process; Hitting time; Left-continuous random walk; Parity; Positive drift; Separable distribution; Skip-free}},
language = {{eng}},
number = {{4}},
publisher = {{Springer}},
series = {{Journal of Theoretical Probability}},
title = {{Left-Continuous Random Walk on Z and the Parity of Its Hitting Times}},
url = {{http://dx.doi.org/10.1007/s10959-025-01440-x}},
doi = {{10.1007/s10959-025-01440-x}},
volume = {{38}},
year = {{2025}},
}