Amnesic Elephant Random Walk with Polynomially Decaying Step Sizes
Abstract
In this paper, we introduce an amnesic elephant random walk with polynomially decaying step sizes. The walk retains the loss of memory effect of amnesic ERW, however its step sizes decay polynomially over time. We study the effect of step-size exponent and memory parameter on the long-time behaviour of the walk. Two critical thresholds are identified that determine its phase diagram. We obtain almost sure convergence results, law of iterated logarithm, asymptotic normality and mean square displacement rate of the walk across different parameter regimes. The polynomial decay of step sizes gives rise to a subdiffusive regime while classical diffusion is recovered in the absence of decay. Also, we identify localization regimes in which the walk converges almost surely to a finite random variable.
Related
- Limit theorems for amnesic elephant Random walks with random increment sizes
- Cramér-type moderate deviations and precise large deviations for the elephant random walk with random step sizes
- Normal approximation, moderate and large deviations for the elephant random walk with stops and random step sizes
- Precise large deviations for the elephant random walk with Weibull-type and lognormal-type random step sizes
- Moments approach for the elephant random walk
BibTeX
@misc{ghosh2026amnesic,
title = {{Amnesic Elephant Random Walk with Polynomially Decaying Step Sizes}},
author = {Shyan Ghosh and Manisha Dhillon and Kuldeep Kumar Kataria},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16956},
eprint = {2609.16956},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16956},
}
RIS
TY - UNPB
TI - Amnesic Elephant Random Walk with Polynomially Decaying Step Sizes
AU - Shyan Ghosh
AU - Manisha Dhillon
AU - Kuldeep Kumar Kataria
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16956
UR - https://arxiv.org/abs/2609.16956
AB - In this paper, we introduce an amnesic elephant random walk with polynomially decaying step sizes. The walk retains the loss of memory effect of amnesic ERW, however its step sizes decay polynomially over time. We study the effect of step-size exponent and memory parameter on the long-time behaviour of the walk. Two critical thresholds are identified that determine its phase diagram. We obtain almost sure convergence results, law of iterated logarithm, asymptotic normality and mean square displacement rate of the walk across different parameter regimes. The polynomial decay of step sizes gives rise to a subdiffusive regime while classical diffusion is recovered in the absence of decay. Also, we identify localization regimes in which the walk converges almost surely to a finite random variable.
ER -