New insights on the minimal random walk

TL;DR

This paper analyzes the asymptotic behavior of the minimal random walk using a new martingale approach, revealing detailed results across its three regimes, including laws of large numbers and Gaussian fluctuations.

Contribution

It introduces a novel martingale method to study the MRW, providing new almost sure asymptotic results and convergence properties across different regimes.

Findings

Established quadratic strong law in diffusive regime

Proved convergence to a nondegenerate limit in superdiffusive regime

Showed Gaussian fluctuations around the limiting variable

Abstract

The aim of this paper is to deepen the analysis of the asymptotic behavior of the so-called minimal random walk (MRW) using a new martingale approach. The MRW is a discrete-time random walk with infinite memory that has three regimes depending on the location of its two parameters. In the diffusive and critical regimes, we establish new results on the almost sure asymptotic behavior of the MRW, such as the quadratic strong law and the law of the iterated logarithm. In the superdiffusive regime, we prove the almost sure convergence of the MRW, properly normalized, to a nondegenerate random variable. Moreover, we show that the fluctuation of the MRW around its limiting random variable is still Gaussian.