A strong invariance principle for the elephant random walk

Author(s)

Coletti, Cristian F. ; Gava, Renato ; Schutz, Gunter M.

• Document type: Journal article
• Source: JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT; v. N/A, p. 8-pg., 2017-12-01.

Abstract

We consider a non-Markovian discrete-time random walk on 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. (AU)

• FAPESP's process: 15/20110-0 - Branching Random Walks and Interacting particle System in Random Environment.
• Grantee: Cristian Favio Coletti
• Support Opportunities: Scholarships abroad - Research
• FAPESP's process: 16/11648-0 - Limit theorems and phase transition results for information propagation models on graphs
• Grantee: Pablo Martin Rodriguez
• Support Opportunities: Regular Research Grants