2017/07/21 by Cristian F. Coletti, Renato Jacob Gava, Renato Gava +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #Brownian motion #Central limit theorem #Diffusion and Search Dynamics #Discrete mathematics #Geometry #Invariance principle #Iterated logarithm #Law of the iterated logarithm #Logarithm #Markov Chains and Monte Carlo Methods #Mathematical analysis #Mathematics #Physics #Random walk #Scaling #Scaling limit #Statistical physics #Statistics #Stochastic processes and statistical mechanics #math.PR
paper · pdf · doi:10.1088/1742-5468/aa9680
published as J. Stat. Mech. (2017) 123207
arxiv created 2017/07/21 · openalex publication_date 2017/12/01 · arxiv updated 2017/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract We consider a non-Markovian discrete-time random walk on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mstyle> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>4</mml:mn> </mml:mstyle> </mml:math> 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.