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Phase transitions for a unidirectional elephant random walk with a power law memory II: Some sharper estimates

Rahul Roy, Masato Takei, Hideki Tanemura

TL;DR

This paper analyzes the unidirectional elephant random walk with a power-law memory exponent $\beta$ in the subcritical range $(-1, p/(1-p))$. It develops a modified process and a coupling argument to obtain sharp asymptotics for the position $S_n$ and to establish positivity of the martingale limit $M_\infty$, which underpins a nondegenerate phase transition. It proves that, for $-1<\beta<0$, $S_n$ scales as $n^{p(\beta+1)-\beta}$ with a nontrivial leading term, while for $0<\beta<p/(1-p)$ the event $\{S_n\to \infty\}$ has probability in $(0,1)$ and, conditional on that event, the same growth rate governs $S_n$; a central limit theorem with a random normalizing factor is obtained in all regimes via a stable martingale CLT. These results complete the phase-transition picture of uERW with power-law memory and provide sharp estimates for the growth of $S_n$ and its fluctuations.

Abstract

We continue our study of the unidirectional elephant random walk (uERW) initiated in {\it {Electron. Commun. Probab.}} ({\bf 29} 2024, article no. 78). In this paper we obtain definitive results when the memory exponent $β\in (-1, p/(1-p))$. In particular using a coupling argument we obtain the exact asymptotic rate of growth of $S_n$, the location of the uERW at time $n$, for the case $β\in (-1, 0] $. Also, for the case $β\in (0, p/(1-p))$ we show that $P(S_n \to \infty) \in (0,1)$ and conditional on $\{S_n \to \infty\}$ we obtain the exact asymptotic rate of growth of $S_n$. In addition we obtain the central limit theorem for $S_n$ when $β\in (-1, p/(1-p))$.

Phase transitions for a unidirectional elephant random walk with a power law memory II: Some sharper estimates

TL;DR

This paper analyzes the unidirectional elephant random walk with a power-law memory exponent in the subcritical range . It develops a modified process and a coupling argument to obtain sharp asymptotics for the position and to establish positivity of the martingale limit , which underpins a nondegenerate phase transition. It proves that, for , scales as with a nontrivial leading term, while for the event has probability in and, conditional on that event, the same growth rate governs ; a central limit theorem with a random normalizing factor is obtained in all regimes via a stable martingale CLT. These results complete the phase-transition picture of uERW with power-law memory and provide sharp estimates for the growth of and its fluctuations.

Abstract

We continue our study of the unidirectional elephant random walk (uERW) initiated in {\it {Electron. Commun. Probab.}} ({\bf 29} 2024, article no. 78). In this paper we obtain definitive results when the memory exponent . In particular using a coupling argument we obtain the exact asymptotic rate of growth of , the location of the uERW at time , for the case . Also, for the case we show that and conditional on we obtain the exact asymptotic rate of growth of . In addition we obtain the central limit theorem for when .

Paper Structure

This paper contains 4 sections, 6 theorems, 78 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A modified version of the process
  3. Positivity of the martingale limit
  4. Central limit theorems

Key Result

Theorem 1.1

Assume that $p \in (0,1)$ and let $N \stackrel{d}{=} N(0,1)$. (i) If $\beta \in (-1,p/(1-p))$ then $\dfrac{W_n}{\sqrt{n^{p(\beta+1)-\beta}}} \stackrel{d}{\to} \eta\cdot N$ as $n\to\infty$, where $N$ is independent of $\eta$. (ii) If $\beta \in (-1, 0]$ then $P(\eta >0) =1$ and $\dfrac{W_n}{\eta \sq

Figures (2)

  • Figure 1: $\eta^{(m)}$ denotes all the integer points on the line $\{t=m\}$ which are eventually connected to the vertex $1$ on the $x$-axis via integer points on the levels $\{1 \leq t \leq m-1\}$. The black lines denote those which do not have any connection from $\{t \geq m\}$.
  • Figure 2: The first three steps of the algorithm.

Theorems & Definitions (10)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3: the Paley--Zygmund inequality
  • Lemma 2.4
  • proof
  • Theorem : Häusler and Luschgy