A strong invariance principle for the elephant random walk. (14th December 2017)

• Record Type: Journal Article
• Authors: Coletti, Cristian F; Gava, Renato; Schütz, Gunter M
• Is Part Of: Journal of statistical mechanics. (2017:Dec.)
• Publication Date: 2017-12-14
• Subjects: 4; Statistical mechanics -- Periodicals; Mechanics -- Statistical methods -- Periodicals; 530.1305
• DOI: 10.1088/1742-5468/aa9680
• Languages: English
• ISSNs: 1742-5468

Abstract

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.