Limit theorems for amnesic elephant Random walks with random increment sizes
Abstract
We introduce a generalized amnesic elephant random walk in which, at each time, the direction of an underlying amnesic elephant random walk selects one of two increment distributions, while the actual increment is sampled from the corresponding independent sequence. The selected distribution does not need to determine the sign of the increment. This construction combines memory reinforcement, amnesia, and randomness in jump sizes. Assuming finite second moments, we establish central limit theorems and functional limit theorems, identifying diffusive, critical, and superdiffusive regimes determined by the memory and amnesia parameters. We further prove a Gaussian fluctuation theorem around the random superdiffusive limit, with an explicit limiting variance that captures the contributions of both the underlying amnesic walk and the random jump magnitudes. For the special case without amnesia, we also establish stable limit theorems and their functional counterparts when the increments belong to the domain of attraction of a non-Gaussian stable law, in the regime where heavy-tailed jumps dominate the memory contribution.
Related
- Optimal Rates in Local and Central Limit Theorems for the Elephant Random Walk
- Amnesic Elephant Random Walk with Polynomially Decaying Step Sizes
- A Superdiffusive Local Limit Theorem for the Elephant Random Walk and the Breakdown of Log-Concavity
- Functional limit theorems for perturbed random walks
- Limit theorems for random matrix products with a simple top Lyapunov exponent
BibTeX
@misc{coletti2026limit,
title = {{Limit theorems for amnesic elephant Random walks with random increment sizes}},
author = {Cristian F. Coletti and Rafael Souza Dos Santos and Glauco Valle},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14871},
eprint = {2609.14871},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14871},
}
RIS
TY - UNPB
TI - Limit theorems for amnesic elephant Random walks with random increment sizes
AU - Cristian F. Coletti
AU - Rafael Souza Dos Santos
AU - Glauco Valle
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.14871
UR - https://arxiv.org/abs/2609.14871
AB - We introduce a generalized amnesic elephant random walk in which, at each time, the direction of an underlying amnesic elephant random walk selects one of two increment distributions, while the actual increment is sampled from the corresponding independent sequence. The selected distribution does not need to determine the sign of the increment. This construction combines memory reinforcement, amnesia, and randomness in jump sizes. Assuming finite second moments, we establish central limit theorems and functional limit theorems, identifying diffusive, critical, and superdiffusive regimes determined by the memory and amnesia parameters. We further prove a Gaussian fluctuation theorem around the random superdiffusive limit, with an explicit limiting variance that captures the contributions of both the underlying amnesic walk and the random jump magnitudes. For the special case without amnesia, we also establish stable limit theorems and their functional counterparts when the increments belong to the domain of attraction of a non-Gaussian stable law, in the regime where heavy-tailed jumps dominate the memory contribution.
ER -