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Estimates on Escape Times for the Elephant Random Walk

Morgan André, Leonel Zuaznábar

Abstract

We study the gambler's ruin problem for the Elephant Random Walk, focusing on escape time from a symmetric interval of the form $\{-N, \ldots, N\}$. As our main result, we derive tight exponential bounds for the tail of this escape time. We then illustrate the usefulness of such bounds by proving that, in the diffusive regime, the Elephant's average behavior mirrors that of the traditional symmetric random walk: the expected escape time grows quadratically with $N$.

Estimates on Escape Times for the Elephant Random Walk

Abstract

We study the gambler's ruin problem for the Elephant Random Walk, focusing on escape time from a symmetric interval of the form . As our main result, we derive tight exponential bounds for the tail of this escape time. We then illustrate the usefulness of such bounds by proving that, in the diffusive regime, the Elephant's average behavior mirrors that of the traditional symmetric random walk: the expected escape time grows quadratically with .
Paper Structure (7 sections, 3 theorems, 32 equations)

This paper contains 7 sections, 3 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Markovian description of the ERW
  3. The Elephant's tail
  4. Proof of the upper bound in Theorem \ref{['main_thm']}
  5. Proof of the lower bound in Theorem \ref{['main_thm']}
  6. Expected duration of Elephant's Ruin
  7. Proof of Lemma \ref{['lemma:sym']}

Key Result

Proposition 1

If $0<p<3/4$ then there exists a constant $\theta$ such that:

Theorems & Definitions (6)

  • Proposition 1
  • Theorem 1
  • Remark
  • Lemma 1
  • proof : Proof of Proposition \ref{['prop_1']}
  • proof : Proof of Lemma \ref{['lemma:sym']}