Limit theorems for a resonant interacting elephant random walk with random amplitudes

Rafik Aguech; Mohamed Abdelkader; Rafik Aguech; Mohamed Abdelkader

doi:10.3934/math.20261379

AIMS Mathematics

2026, Volume 11, Issue 10: 35551-35577. doi: 10.3934/math.20261379

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Research article

Limit theorems for a resonant interacting elephant random walk with random amplitudes

Rafik Aguech ^,,
Mohamed Abdelkader

Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 29 August 2026 Revised: 15 October 2026 Accepted: 21 October 2026 Published: 29 October 2026
MSC : 60F05, 60F15, 60G42, 60G50, 60K35, 82C41

We study two interacting elephant random walks. The second has memory $ \alpha_2 = 2p_2-1\in(-1, 1] $: It repeats a uniformly sampled past step with probability $ p_2 $ and reverses it otherwise. With probability $ \beta $, the first copies a past step of the second, multiplies it by an independent amplitude of mean $ m_L $ and second moment $ v $, and repeats or reverses it with memory $ \alpha_1 = 2p_1-1 $; otherwise, it does the same with its own past and memory $ \alpha_3 = 2p_3-1 $. Its effective memory is $ \gamma = \alpha_3(1-\beta) $, and the mean coupling is $ d = \beta\alpha_1 m_L $. In the resonant regime $ \gamma = \alpha_2 $, and under the moment assumptions stated in Theorem 3.1, the first position is driven by the source $ d\, S_n^{(2)}/n $. If $ \gamma > 1/2 $ and $ d\neq 0 $, $ S_n^{(1)}/(n^{\gamma}\log n) $ converges almost surely to $ d\, L $, where $ L $ is the limit of the second walk. If $ \gamma = 1/2 $ and $ d\neq 0 $, $ S_n^{(1)}/\sqrt{n(\log n)^{3}} $ converges in law to $ \mathcal N(0, d^{2}/3) $. If $ -1 < \gamma < 1/2 $ and $ \mathbb{E}[|L_1|^{2+\delta}] < \infty $ for some $ \delta > 0 $, the $ \sqrt n $ fluctuations are jointly Gaussian, and the variance of the first coordinate is $ v/(1-2\gamma)+2d^{2}/(1-2\gamma)^{3} $. If $ d = 0 $ and $ \gamma = 1/2 $, under a moment of order $ 2+\delta $, the scale returns to $ \sqrt{n\log n} $, with variance $ v $. The case $ \gamma\neq\alpha_2 $ is left to a companion manuscript, which is not used in the proofs.
- elephant random walk,
- interacting random walks,
- random amplitudes,
- resonance,
- Jordan block,
- martingales,
- central limit theorem,
- law of the iterated logarithm
Citation: Rafik Aguech, Mohamed Abdelkader. Limit theorems for a resonant interacting elephant random walk with random amplitudes[J]. AIMS Mathematics, 2026, 11(10): 35551-35577. doi: 10.3934/math.20261379

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Abstract

We study two interacting elephant random walks. The second has memory $ \alpha_2 = 2p_2-1\in(-1, 1] $: It repeats a uniformly sampled past step with probability $ p_2 $ and reverses it otherwise. With probability $ \beta $, the first copies a past step of the second, multiplies it by an independent amplitude of mean $ m_L $ and second moment $ v $, and repeats or reverses it with memory $ \alpha_1 = 2p_1-1 $; otherwise, it does the same with its own past and memory $ \alpha_3 = 2p_3-1 $. Its effective memory is $ \gamma = \alpha_3(1-\beta) $, and the mean coupling is $ d = \beta\alpha_1 m_L $. In the resonant regime $ \gamma = \alpha_2 $, and under the moment assumptions stated in Theorem 3.1, the first position is driven by the source $ d\, S_n^{(2)}/n $. If $ \gamma > 1/2 $ and $ d\neq 0 $, $ S_n^{(1)}/(n^{\gamma}\log n) $ converges almost surely to $ d\, L $, where $ L $ is the limit of the second walk. If $ \gamma = 1/2 $ and $ d\neq 0 $, $ S_n^{(1)}/\sqrt{n(\log n)^{3}} $ converges in law to $ \mathcal N(0, d^{2}/3) $. If $ -1 < \gamma < 1/2 $ and $ \mathbb{E}[|L_1|^{2+\delta}] < \infty $ for some $ \delta > 0 $, the $ \sqrt n $ fluctuations are jointly Gaussian, and the variance of the first coordinate is $ v/(1-2\gamma)+2d^{2}/(1-2\gamma)^{3} $. If $ d = 0 $ and $ \gamma = 1/2 $, under a moment of order $ 2+\delta $, the scale returns to $ \sqrt{n\log n} $, with variance $ v $. The case $ \gamma\neq\alpha_2 $ is left to a companion manuscript, which is not used in the proofs.

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