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A note on the long time behavior of the elephant random walk with stops

Tatsuya Akimoto, Masato Takei, Keisuke Taniguchi

Abstract

We study the long time behavior of the elephant random walk with stops, introduced by Kumar, Harbola and Lindenberg (2010), and establish the phase transition of the number of visited points up to time $n$, and the correlation between the position at time $n$ and the number of moves up to time $n$.

A note on the long time behavior of the elephant random walk with stops

Abstract

We study the long time behavior of the elephant random walk with stops, introduced by Kumar, Harbola and Lindenberg (2010), and establish the phase transition of the number of visited points up to time , and the correlation between the position at time and the number of moves up to time .

Paper Structure

This paper contains 7 sections, 10 theorems, 32 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Almost sure range of the elephant random walk with stops
  4. Correlation between the position and the number of moves
  5. Proof of Theorem \ref{['thm:asrangeERW']}
  6. Proof of Theorem \ref{['thm:Taniguchi23ThmA']}
  7. Proof of Theorem \ref{['thm:Taniguchi23ThmB']}

Key Result

Theorem 1.1

Let $\phi(x):= \sqrt{2x\log \log x}$. (i) (Subcritical regime) If $a \in [-b,b/2)$ then $\dfrac{S_n}{\sqrt{\Sigma_n}}\stackrel{d}{\to}N\left(0,\dfrac{b}{b-2a}\right)$ as $n \to \infty$, and $\limsup_{n \to \infty} \pm \frac{S_n}{\phi(\Sigma_n)}=\sqrt{\dfrac{b}{b-2a}}$ a.s.. (ii) (Critical regime) If

Figures (1)

  • Figure 1: The graphs of $\rho=\lim_{n \to \infty} \rho[S_n^2,\Sigma_n]$ as a function of $a \in [-b,b]$ for $b=0.3$, $0.6$ and $0.9$. Note that the critical value is $a_c(b)=b/2$.

Theorems & Definitions (17)

  • Theorem 1.1: Bercu Bercu22
  • Remark 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 7 more