Elephant Random Walk Conditioned to Avoid Zero

2026/07/30 by Kohki Iba, Go Tokumitsu
Mathematics · #math.PR

paper · pdf

37 pages, 1 figure

arxiv created 2026/07/30 · arxiv updated 2026/08/03

Abstract

We investigate the long-time limit of an elephant random walk conditioned to avoid zero and show that the resulting process admits a Skorokhod-type embedding into a three-dimensional Bessel process. Using this embedding, we further establish several limit theorems for the conditioned elephant random walk, including a scaling limit to a deterministically rescaled and time-changed three-dimensional Bessel process, and a law of the iterated logarithm.

Citations

• How often does a critical elephant random walk return to origin
• On the local times of noise reinforced Bessel processes
• On universality in penalisation problems with multiplicative weights
• Counting the zeros of an elephant random walk
• Asymptotic Analysis of the Elephant Random Walk