The Shark Random Swim (Lévy flight with memory)

Abstract

The Elephant Random Walk (ERW), first introduced by Schütz and Trimper (2004), is a one-dimensional simple random walk on $\mathbb{Z}$ having a memory about the whole past. We study the Shark Random Swim, a random walk whose steps are $\alpha$-stable distributed with memory about the whole past. In contrast with the ERW, the steps of the Shark Random Swim have a heavy tailed distribution. Our aim in this work is to study the impact of the heavy tailed step distributions on the asymptotic behavior of the random walk. We shall see that, as for the ERW, the asymptotic behavior of the Shark Random Swim depends on its memory parameter $p$, and that a phase transition can be observed at the critical value $p=\frac{1}{\alpha }$.

Added an extension to convergence of the finite dimensional distributions and corrected a mistake in Lemma 11

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Cited by in corpus (12)

• Gaussian fluctuation for superdiffusive elephant random walks
• Random walks with preferential relocations and fading memory: a study through random recursive trees
• 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
• Multidimensional walks with random tendency
• The diffusion of opposite opinions in a randomly biased environment
• Reinforced random walks under memory lapses
• Recurrence and transience of multidimensional elephant random walks
• Broadcasting induced colourings of random recursive trees and preferential attachment trees
• Ulam's History-dependent Random Adding Process