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A universal scaling limit for diffusive amnesic step-reinforced random walks

Marco Bertenghi, Lucile Laulin

Abstract

We introduce a variation of the step-reinforced random walk with general memory. For the diffusive regime, we establish a functional invariance principle and show that, given suitable conditions on the memory sequence, the arising limiting processes are always the sum of a noise reinforced Brownian motion and a (not independent) Brownian motion.

A universal scaling limit for diffusive amnesic step-reinforced random walks

Abstract

We introduce a variation of the step-reinforced random walk with general memory. For the diffusive regime, we establish a functional invariance principle and show that, given suitable conditions on the memory sequence, the arising limiting processes are always the sum of a noise reinforced Brownian motion and a (not independent) Brownian motion.
Paper Structure (13 sections, 23 theorems, 102 equations, 1 table)

This paper contains 13 sections, 23 theorems, 102 equations, 1 table.

Table of Contents

  1. Introduction
  2. The Model
  3. Main results
  4. Law of large numbers
  5. A functional invariance principle
  6. Regularly varying sequences
  7. A two dimensional martingale approach
  8. Proof of main results
  9. Proof of Theorem \ref{['thm:LLN']}
  10. Proof of Theorem \ref{['thm:MainResult']}
  11. Technical Lemmas
  12. Truncation argument for removing the boundness assumption
  13. A non-standard result on martingales

Key Result

Theorem 3.1

For $\frac{\alpha-1}{\alpha} <p < \frac{2 \alpha-1}{2 \alpha}$ we have the almost sure convergence

Theorems & Definitions (42)

  • Theorem 3.1: Strong law of large numbers
  • Theorem 3.2
  • Remark 3.3
  • Definition 4.1: Slowly varying function
  • Definition 4.2: Regularly varying function
  • Definition 4.3: Regularly varying sequence
  • Theorem 4.4: See Display (1.1) and Theorem 6 in Bojanic73
  • Theorem 4.5: Theorem 4 in Bojanic73
  • Corollary 4.6
  • Proposition 4.7
  • ...and 32 more