On the asymptotics of a lazy reinforced random walk

Based on a martingale theory approach, we present a complete characterization of the asymptotic behaviour of a lazy reinforced random walk (LRRW) which shows three different regimes (diffusive, critical and superdiffusive). This allows us to prove versions of the law of large numbers, the quadratic...

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Bibliographic Details
Published in:arXiv.org 2024-02
Main Authors: González-Navarrete, Manuel, Lambert, Rodrigo, Vázquez Guevara, Víctor Hugo
Format: Article
Language:English
Subjects:
Asymptotic properties
Logarithms
Martingales
Random variables
Random walk
Theorems
Online Access:Get full text
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Summary:Based on a martingale theory approach, we present a complete characterization of the asymptotic behaviour of a lazy reinforced random walk (LRRW) which shows three different regimes (diffusive, critical and superdiffusive). This allows us to prove versions of the law of large numbers, the quadratic strong law, the law of iterated logarithm, the almost sure central limit theorem and the functional central limit theorem in the diffusive and critical regimes. In the superdiffusive regime we obtain a strong convergence to a random variable, including a central limit theorem and a law of iterated logarithm for the fluctuations.
ISSN:2331-8422