Limit theorems for a minimal random walk model

Abstract

We study the minimal random walk introduced by Kumar, Harbola and Lindenberg. It is a random process on $\{0,1,\dots \}$ 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.

arXiv commentPublished at https://iopscience.iop.org/article/10.1088/1742-5468/ab3343 in the Journal of Statistical Mechanics

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• Limit theorems for the 'laziest' minimal random walk model of elephant type
• New insights on the minimal random walk
• Limit theorems for a random walk with memory perturbed by a dynamical system