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On the asymptotics of a lazy reinforced random walk

Manuel González-Navarrete, Rodrigo Lambert, Víctor Hugo Vázquez Guevara

Abstract

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.

On the asymptotics of a lazy reinforced random walk

Abstract

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.
Paper Structure (23 sections, 15 theorems, 96 equations)

This paper contains 23 sections, 15 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. The Random Walk
  3. Main Results
  4. The diffusive regime
  5. The critical regime
  6. The superdiffusive regime
  7. Demonstrations of achieved results
  8. Martingale Analysis
  9. The diffusive regime
  10. Proof of theorem \ref{['CVG1']}
  11. Proof of Theorem \ref{['thmmoments']}
  12. Proof of Theorem \ref{['CVG3']}
  13. Proof of Theorem \ref{['thmASCLT']}
  14. Proof of Theorem \ref{['T-FCLT-DR']}
  15. The critical regime
  16. ...and 8 more sections

Key Result

Theorem 1

If $\alpha<1/2$, then we have the following almost sure convergence to be precise

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 5 more