Limit theorems for a minimal random walk model

TL;DR

This paper analyzes a minimal random walk with unbounded memory, establishing fundamental limit theorems across different regimes, including convergence to a non-normal distribution in a mixed regime scenario.

Contribution

It proves the law of large numbers, central limit theorem, and law of the iterated logarithm for the minimal random walk, including a novel convergence result in a three-regime case.

Findings

Law of large numbers for all parameters

Central limit theorem under diffusive regimes

Convergence to a non-normal distribution in mixed regimes

Abstract

We study the minimal random walk introduced by Kumar, Harbola and Lindenberg. It is a random process on $\{0, 1, \ldots \}$ with unbounded memory which exhibits subdiffusive, diffusive and superdiffusive regimes. We prove the law of large numbers for the whole parameter set. Then we prove the central limit theorem and the law of the iterated logarithm for the minimal random walk under diffusive and marginally superdiffusive behaviors. More interestingly, we establish a result for the minimal random walk when it possesses the three regimes; we show the convergence of its rescaled version to a non-normal random variable.