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Functional limit theorems for elephant random walks on general periodic structures

Shuhei Shibata

TL;DR

This work extends functional limit theorems for Elephant Random Walks to general periodic state spaces Γ, including lattices like triangular, hexagonal, and brick-wall structures. Using a Polya-type urn representation, the authors reduce ERW dynamics to urn processes governed by replacement matrices with eigenstructure that dictates phase transitions at critical memory p_c. They prove LLN and functional limit theorems across diffusive, critical, and superdiffusive regimes, revealing structure-dependent quantities that alter covariance, scaling, and limiting distributions compared to the classical Z^d setting. The results are complemented by explicit parameterizations for common lattices, illustrating how the lattice geometry fundamentally influences asymptotic behavior and revealing new phenomena absent in standard ERW on Z^d.

Abstract

This paper investigates functional limit theorems for the Elephant Random Walk (ERW) on general periodic structures, extending the Bertenghi's results on $\mathbb{Z}^d$. Our results reveal new structure-dependent quantities that do not appear in the classical setting $\mathbb{Z}^d$, highlighting how the underlying structure affects the asymptotic behavior of the walk.

Functional limit theorems for elephant random walks on general periodic structures

TL;DR

This work extends functional limit theorems for Elephant Random Walks to general periodic state spaces Γ, including lattices like triangular, hexagonal, and brick-wall structures. Using a Polya-type urn representation, the authors reduce ERW dynamics to urn processes governed by replacement matrices with eigenstructure that dictates phase transitions at critical memory p_c. They prove LLN and functional limit theorems across diffusive, critical, and superdiffusive regimes, revealing structure-dependent quantities that alter covariance, scaling, and limiting distributions compared to the classical Z^d setting. The results are complemented by explicit parameterizations for common lattices, illustrating how the lattice geometry fundamentally influences asymptotic behavior and revealing new phenomena absent in standard ERW on Z^d.

Abstract

This paper investigates functional limit theorems for the Elephant Random Walk (ERW) on general periodic structures, extending the Bertenghi's results on . Our results reveal new structure-dependent quantities that do not appear in the classical setting , highlighting how the underlying structure affects the asymptotic behavior of the walk.

Paper Structure

This paper contains 8 sections, 4 theorems, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Settings
  3. Definition of the ERW on Gamma
  4. Connection to the Pólya-type urns
  5. Main result
  6. A LLN-type convergence result
  7. Functional-type convergence results
  8. Key quantities for typical examples

Key Result

Theorem 5.1

Let $\{S_n\}_{n=0}^\infty$ be Type-I or Type-II ERW. Then, for all $p\in(0,1)$, we have the almost sure convergence where $\Bar{u}=\frac{1}{m}\sum_{i=1}^m u_i$ and $\Bar{w}=\frac{1}{m'}\sum_{j=1}^{m'} w_j$.

Figures (2)

  • Figure 1: Figure $1$. Left: Triangular lattice. All vertices (black dots) are structurally equivalent. Center: Hexagonal lattice. The vertices are partitioned into two structurally distinct classes, represented by red and blue dots. Right: Brick wall. As in the hexagonal lattice, the vertices are divided into two structurally distinct classes (red and blue dots).
  • Figure 2: Figure $2$. Kagome lattice. The vertices are partitioned into three structurally distinct classes, represented by red, blue, and green dots.

Theorems & Definitions (19)

  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Remark 4.1
  • Remark 4.2
  • Theorem 5.1
  • proof
  • Remark 5.2
  • Theorem 5.3
  • Remark 5.4
  • ...and 9 more