Limit theorems for the fluctuation of the dynamic elephant random walk in the superdiffusive case

TL;DR

The paper analyzes the dynamic elephant random walk in the superdiffusive regime ($p>3/4$) with a strong elephant component, deriving fluctuation limits without assuming coefficient convergence. By recasting the process in a martingale framework and subtracting a random drift, the authors establish a Central Limit Theorem and a Law of the Iterated Logarithm for the scaled fluctuations, with normal limit variance determined by $A_ty^2-A_n^2$ and a drift term $a_nM$. Key steps include detailed summability and asymptotic analyses of the sequences $a_n$, $A_n^2$, and $B_n^2$, and the construction of auxiliary martingales to verify Lindeberg-type conditions. The results extend prior work by relaxing coefficient-convergence requirements while clarifying the role of the random drift in the strong elephant regime, thereby advancing the theoretical understanding of memory-driven, non-Markovian random walks in the superdiffusive phase.

Abstract

Motivated by the previous results by Coletti-de Lima-Gava-Luiz (2020) and Shiozawa (2022), we study the fluctuation of the dynamic elephant random walk in the superdiffusive case with a strong elephant component. Applying the martingale convergence theorem, we prove the Central Limit Theorem and the Law of Iterated Logarithm, where a random drift is subtracted from the process considered.

Theorems & Definitions (12)

• Theorem 1.1: Coletti et al. Col
• Theorem 1.2: Coletti et al. Col
• Theorem 1.3
• Theorem 1.4: Kubota--Takei KaT
• Theorem 1.5: Shiozawa Sio
• Lemma 2.1: Lemma 10 of Col
• Lemma 2.2: Lemma 12 of Col
• Lemma 2.3
• proof
• Lemma 2.4