A strong invariance principle for the elephant random walk

Abstract

We consider a non-Markovian discrete-time random walk on $\mathbb{Z}$ with unbounded memory called the elephant random walk (ERW). We prove a strong invariance principle for the ERW. More specifically, we prove that, under a suitable scaling and in the diffusive regime as well as at the critical value $p_c=3/4$ where the model is marginally superdiffusive, the ERW is almost surely well approximated by a Brownian motion. As a by-product of our result we get the law of iterated logarithm and the central limit theorem for the ERW.

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• On a class of random walks with reinforced memory
• On the elephant random walk with stops playing hide and seek with the Mittag-Leffler distribution
• Limit theorems for the 'laziest' minimal random walk model of elephant type
• Joint invariance principles for random walks with positively and negatively reinforced steps
• On Wasserstein-1 distance in the central limit theorem for elephant random walk
• Multi-Dimensional Elephant Random Walk with Coupled Memory
• New insights on the minimal random walk
• Rate of moment convergence in the central limit theorem for the elephant random walk
• Recurrence and transience of multidimensional elephant random walks
• Asymptotic Normality of Superdiffusive Step-Reinforced Random Walks
• The elephant random walk in the triangular array setting
• Cramér moderate deviations for the elephant random walk
• Limit theorems for a random walk with memory perturbed by a dynamical system