A Superdiffusive Local Limit Theorem for the Elephant Random Walk and the Breakdown of Log-Concavity
Abstract
Let (Sn)n≥ 1 be the one-dimensional Elephant Random Walk (ERW) in the superdiffusive regime, with memory parameter p∈(3/4,1) and first-step bias ℙ(S1=1)=q. Set a=2p−1∈(1/2,1). If fq,a denotes the density of the superdiffusive limit Lq,a, we prove the uniform local limit theorem limn→∞supj∈ℤ, j≡ n (mod 2)|na/2ℙ(Sn=j)−fq,a(j/na)|=0. The proof uses the exact recurrence for the probability mass function of Sn, yielding uniform O(n−a) and O(n−2a) bounds for the p.m.f. and its first discrete differences, respectively. We then disprove a conjecture of Guérin, Laulin, Raschel and Simon concerning eventual log-concavity. Writing La=L1,a and fa=f1,a, let aev be the eventual-log-concavity threshold and let a⋆ be the upper failure threshold for log-concavity of fa. Their results imply aev≥(√5−1)/2, and we prove (√5−1)/2≤ aev≤min{0.80399,a⋆}≤ a⋆<0.918. The bound a⋆<0.918 follows from the first three exact moments of La and the sharp skewness inequality for centered log-concave distributions, while aev≤0.80399 follows from a certified finite-order analysis of the exact p.m.f. recurrence. Direct numerical iterations provide evidence that aev≈0.80399, equivalently pev≈0.902.
Related
- Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift
- The Prodigal Elephant Returns: Last-Return Laws in the Superdiffusive Regime
- Superdiffusive central limit theorem for a class of driven diffusive systems at the critical dimension
- Cramér-type moderate deviations and precise large deviations for the elephant random walk with random step sizes
- Moments approach for the elephant random walk
BibTeX
@misc{trinas2026superdiffusive,
title = {{A Superdiffusive Local Limit Theorem for the Elephant Random Walk and the Breakdown of Log-Concavity}},
author = {Hélio Trinas and Glauco Valle},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14753},
eprint = {2609.14753},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14753},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Superdiffusive Local Limit Theorem for the Elephant Random Walk and the Breakdown of Log-Concavity
AU - Hélio Trinas
AU - Glauco Valle
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14753
UR - https://arxiv.org/abs/2609.14753
AB - Let (S_n)_n≥ 1 be the one-dimensional Elephant Random Walk (ERW) in the superdiffusive regime, with memory parameter p∈(3/4,1) and first-step bias ℙ(S_1=1)=q. Set a=2p−1∈(1/2,1). If f_q,a denotes the density of the superdiffusive limit L_q,a, we prove the uniform local limit theorem lim_n→∞sup_j∈ℤ, j≡ n (mod 2)|n^a/2ℙ(S_n=j)−f_q,a(j/n^a)|=0. The proof uses the exact recurrence for the probability mass function of S_n, yielding uniform O(n^−a) and O(n^−2a) bounds for the p.m.f. and its first discrete differences, respectively. We then disprove a conjecture of Guérin, Laulin, Raschel and Simon concerning eventual log-concavity. Writing L_a=L_1,a and f_a=f_1,a, let a_ev be the eventual-log-concavity threshold and let a_⋆ be the upper failure threshold for log-concavity of f_a. Their results imply a_ev≥(√5−1)/2, and we prove (√5−1)/2≤ a_ev≤min{0.80399,a_⋆}≤ a_⋆<0.918. The bound a_⋆<0.918 follows from the first three exact moments of L_a and the sharp skewness inequality for centered log-concave distributions, while a_ev≤0.80399 follows from a certified finite-order analysis of the exact p.m.f. recurrence. Direct numerical iterations provide evidence that a_ev≈0.80399, equivalently p_ev≈0.902.
ER -