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Stable functional CLTs for scaled elephant random walks

Go Tokumitsu

Abstract

We establish stable functional central limit theorems for scaled elephant random walks in the diffusive, critical, and superdiffusive cases using the martingale approach.

Stable functional CLTs for scaled elephant random walks

Abstract

We establish stable functional central limit theorems for scaled elephant random walks in the diffusive, critical, and superdiffusive cases using the martingale approach.
Paper Structure (11 sections, 13 theorems, 66 equations)

This paper contains 11 sections, 13 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Previous results
  4. Organization of this paper
  5. Proofs of main results
  6. Superdiffusive case
  7. Diffusive case
  8. Critical case
  9. Appendix
  10. Stable martingale FCLT
  11. Technical lemmas

Key Result

Theorem 1.1

Let $(S_n)$ be the ERW with $3/4<p<1$. Then the following joint convergence in $D\times \mathcal{Y}$ holds: where $Y$ is an arbitrary $\mathcal{F}$-measurable random variable taking values in a separable metrizable space $\mathcal{Y}$ and where $W=(W(t))_{t\ge0}$ is a standard Brownian motion independent of $Y$. In particular, we may take $Y=L_{p,q}$, where $L_{p,q}$ denotes the almost sure limit

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8: Coletti--Gava--Schütz Col2
  • Theorem 1.9: Coletti--Gava--Schütz Col2
  • Theorem 1.10: Kubota--Takei KaT
  • ...and 15 more