Functional central limit theorems for the dynamic elephant random walk

TL;DR

This paper establishes functional central limit theorems for the dynamic elephant random walk, revealing its asymptotic behavior in specific scaling regimes using advanced probabilistic techniques.

Contribution

It introduces new functional CLTs for the dynamic elephant random walk, extending understanding of its long-term behavior under different scaling orders.

Findings

Proves CLTs in  and  log n regimes.

Uses martingale convergence and Karamata's regular variation.

Provides theoretical foundation for the asymptotic analysis of the process.

Abstract

We prove functional central limit theorems for the dynamic elephant random walk in the $\sqrt{n}$ and $\sqrt{n\log n}$ orders, by applying the martingale convergence theorem and Karamata's theory of regular variation.