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

Vilkas, Timo LU orcid (2024) p.1-14
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.
Please use this url to cite or link to this publication:
author
organization
alternative title
Vänsterkontinuerlig slumpvandring på Z och pariteten av dess första träfftider
publishing date
type
Working paper/Preprint
publication status
published
subject
keywords
Left-continuous random walk on Z, positive drift, skip-free to the left, hitting time, parity, separable distribution, branching process
pages
14 pages
publisher
arXiv.org
DOI
10.48550/arXiv.2407.06903
language
English
LU publication?
yes
id
877f89b0-4760-4f95-be1d-f9e2ee892fab
date added to LUP
2025-03-10 10:05:24
date last changed
2025-04-04 14:04:06
@misc{877f89b0-4760-4f95-be1d-f9e2ee892fab,
  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.}},
  author       = {{Vilkas, Timo}},
  keywords     = {{Left-continuous random walk on Z; positive drift; skip-free to the left; hitting time; parity; separable distribution; branching process}},
  language     = {{eng}},
  month        = {{07}},
  note         = {{Preprint}},
  pages        = {{1--14}},
  publisher    = {{arXiv.org}},
  title        = {{Left-continuous random walk on Z and the parity of its hitting times}},
  url          = {{http://dx.doi.org/10.48550/arXiv.2407.06903}},
  doi          = {{10.48550/arXiv.2407.06903}},
  year         = {{2024}},
}