Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Phase transitions for a unidirectional elephant random walk with a power law memory

Rahul Roy, Masato Takei, Hideki Tanemura

Abstract

For the standard elephant random walk, Laulin (2022) studied the case when the increment of the random walk is not uniformly distributed over the past history instead has a power law distribution. We study such a problem for the unidirectional elephant random walk introduced by Harbola, Kumar and Lindenberg (2014). Depending on the memory parameter $p$ and the power law exponent $β$, we obtain three distinct phases in one such phase the elephant travels only a finite distance almost surely, and the other two phases are distinguished by the speed at which the elephant travels.

Phase transitions for a unidirectional elephant random walk with a power law memory

Abstract

For the standard elephant random walk, Laulin (2022) studied the case when the increment of the random walk is not uniformly distributed over the past history instead has a power law distribution. We study such a problem for the unidirectional elephant random walk introduced by Harbola, Kumar and Lindenberg (2014). Depending on the memory parameter and the power law exponent , we obtain three distinct phases in one such phase the elephant travels only a finite distance almost surely, and the other two phases are distinguished by the speed at which the elephant travels.
Paper Structure (8 sections, 14 theorems, 135 equations, 1 table)

This paper contains 8 sections, 14 theorems, 135 equations, 1 table.

Table of Contents

  1. Introduction
  2. Results
  3. Proof of Theorem \ref{['thm:EofXi_n']}
  4. Proof of Theorem \ref{['r-thmsmallbeta']}
  5. Proof of Theorem \ref{['r-thm:PositiveBetabdd']}
  6. Proof of Theorem \ref{['r-thm:Extinction']}
  7. Moments for the critical case
  8. Conclusions

Key Result

Theorem 2.1

For $\beta>-1$,

Theorems & Definitions (28)

  • Remark 1.1
  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm:EofXi_n']}
  • Proposition 4.1
  • ...and 18 more