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Moments approach for the elephant random walk
Abstract
We discuss the method of moments for the one-dimensional elephant random walk (ERW). We first derive a differential recurrence relation for the characteristic function of the ERW, which yields a corresponding system of recurrence relations for its moments. We then obtain asymptotic approximations for the moments in each of the three parameter regimes of the ERW. Finally, by establishing the convergence of the moments and verifying the corresponding moment-determinacy conditions, we identify the limiting distributions of the ERW in each regime.
Related
- 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
- A Generalized Elephant Random Walk with Time-Dependent Memory
- 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{vishwakarma2026moments,
title = {{Moments approach for the elephant random walk}},
author = {Pradeep Vishwakarma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11035},
eprint = {2609.11035},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11035},
}
RIS
TY - UNPB
TI - Moments approach for the elephant random walk
AU - Pradeep Vishwakarma
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11035
UR - https://arxiv.org/abs/2609.11035
AB - We discuss the method of moments for the one-dimensional elephant random walk (ERW). We first derive a differential recurrence relation for the characteristic function of the ERW, which yields a corresponding system of recurrence relations for its moments. We then obtain asymptotic approximations for the moments in each of the three parameter regimes of the ERW. Finally, by establishing the convergence of the moments and verifying the corresponding moment-determinacy conditions, we identify the limiting distributions of the ERW in each regime.
ER -