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

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 ${\hat{I}}^2\in (-1,p/(1-p))$. 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 ${\hat{I}}^2\in (-1,0]$. Also, for the case ${\hat{I}}^2\in (0,p/(1-p))$ 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 ${\hat{I}}^2\in (-1,p/(1-p))$.

13 pages, 2 figures