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On a Class of Random Walks with Reinforced MemoryJournal of Statistical Physics
- Publisher
- Springer International Publishing
- Copyright
- © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
- ISBN
- 978-3-030-60753-1
- Pages
- 147 –161
- DOI
- 10.1007/978-3-030-60754-8_7
- Publisher site
- See Chapter on Publisher Site
Abstract
[A noise reinforced Brownian motion is a centered Gaussian process B̂=(B̂(t))t≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat B=(\hat B(t))_{t\geq 0}$$ \end{document} with covariance 𝔼(B̂(t)B̂(s))=(1−2p)−1tps1−pfor0≤s≤t,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle \mathbb {E}(\hat B(t)\hat B(s))=(1-2p)^{-1}t^ps^{1-p} \quad \text{for} \quad 0\leq s \leq t, $$ \end{document} where p ∈ (0, 1∕2) is a reinforcement parameter. Our main purpose is to establish a version of Donsker’s invariance principle for a large family of step-reinforced random walks in the diffusive regime, and more specifically, to show that B̂\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat B$$ \end{document} arises as the universal scaling limit of the former. This extends known results on the asymptotic behavior of the so-called elephant random walk.]
Published: Nov 4, 2020
Keywords: Reinforcement; Brownian motion; Invariance principle; Elephant random walk; 60G50; 60G51; 60K35