PreprintCC BY-NC-ND 4.0arXiv
A Generalized Elephant Random Walk with Time-Dependent Memory
Abstract
The elephant random walk (ERW) is a discrete-time random walk with complete memory. At every step the walker chooses one of the previous increments at random and then either repeats with a constant probability p∈ [0,1] or reverses it. Since its introduction by Schütz and Trimper, the model has become a useful probabilistic framework for studying long-range dependence, anomalous diffusion, reinforcement, and phase transitions. This paper proposes a generalized ERW model with time-dependent memory. In this generalized model, the usual constant memory parameter p in the standard ERW is allowed to change with time. The cental limit theorems, and the law of the iterated logarithm and some other strong laws are obtained.
Related
- Cramér-type moderate deviations and precise large deviations for the elephant random walk with random step sizes
- Moments approach for the elephant random walk
- Normal approximation, moderate and large deviations for the elephant random walk with stops and random step sizes
- Optimal Rates in Local and Central Limit Theorems for the Elephant Random Walk
- Precise large deviations for the elephant random walk with Weibull-type and lognormal-type random step sizes
BibTeX
@misc{fan2026generalized,
title = {{A Generalized Elephant Random Walk with Time-Dependent Memory}},
author = {Xiequan Fan and Yongcheng Qi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16065},
eprint = {2610.16065},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16065},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Generalized Elephant Random Walk with Time-Dependent Memory
AU - Xiequan Fan
AU - Yongcheng Qi
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16065
UR - https://arxiv.org/abs/2610.16065
AB - The elephant random walk (ERW) is a discrete-time random walk with complete memory. At every step the walker chooses one of the previous increments at random and then either repeats with a constant probability p∈ [0,1] or reverses it. Since its introduction by Schütz and Trimper, the model has become a useful probabilistic framework for studying long-range dependence, anomalous diffusion, reinforcement, and phase transitions. This paper proposes a generalized ERW model with time-dependent memory. In this generalized model, the usual constant memory parameter p in the standard ERW is allowed to change with time. The cental limit theorems, and the law of the iterated logarithm and some other strong laws are obtained.
ER -