Functional central limit theorem for dependent models with finite memory

TL;DR

This paper proves a functional central limit theorem for dependent models with finite memory using martingale techniques, analyzing their limiting behavior and connections to reinforced random walks.

Contribution

It introduces a martingale-based approach to establish a functional central limit theorem for models with finite memory, extending existing results.

Findings

Established a functional central limit theorem for dependent models with finite memory

Analyzed the limiting behavior of the center of mass in these models

Connected the results to reinforced random walks in the literature

Abstract

We provide complementary results for a family of models with dependence on their previous $k$-sum. Using a martingale-based approach, we establish a functional central limit theorem and analyze the limiting behavior of the center of mass. Additionally, we explore the connection between our findings and the study of certain reinforced random walks in the literature