PreprintCC BY-NC-ND 4.0arXiv
Precise large deviations for the elephant random walk with Weibull-type and lognormal-type random step sizes
Abstract
We prove precise large deviations for the symmetric elephant random walk with random step sizes. When the distribution of |Z1−1| is subexponential, we first show that, for each fixedp∈[0,1], there exists a positive number sequence bn→∞ such that ℙ(Sn>x)∼ ℙ(Sn<−x)∼ n/2F̅(x) uniformly for x≥ bn, where F̅(x)=ℙ(|Z1−1|>x). This existence result requires no moment assumptions and does not specify the growth rate of bn. For p∈[0,3/4], we further give explicit sufficient conditions on bn under Weibull-type and lognormal-type tail conditions with a regularly varying factor xγL(x), where γ∈R and L is a positive measurable slowly varying function. Analogous results hold with ℙ(|Z1|>x) and non-strict tails.
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BibTeX
@misc{liang2026precise,
title = {{Precise large deviations for the elephant random walk with Weibull-type and lognormal-type random step sizes}},
author = {Shuyuan Liang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01440},
eprint = {2611.01440},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01440},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Precise large deviations for the elephant random walk with Weibull-type and lognormal-type random step sizes
AU - Shuyuan Liang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01440
UR - https://arxiv.org/abs/2611.01440
AB - We prove precise large deviations for the symmetric elephant random walk with random step sizes. When the distribution of |Z_1−1| is subexponential, we first show that, for each fixedp∈[0,1], there exists a positive number sequence b_n→∞ such that ℙ(S_n>x)∼ ℙ(S_n<−x)∼ n/2F̅(x) uniformly for x≥ b_n, where F̅(x)=ℙ(|Z_1−1|>x). This existence result requires no moment assumptions and does not specify the growth rate of b_n. For p∈[0,3/4], we further give explicit sufficient conditions on b_n under Weibull-type and lognormal-type tail conditions with a regularly varying factor x^γL(x), where γ∈R and L is a positive measurable slowly varying function. Analogous results hold with ℙ(|Z_1|>x) and non-strict tails.
ER -