A remark on elephant random walks via the classical law of the iterated logarithm for self-similar Gaussian processes

Abstract: This paper investigates whether two independent Elephant Random Walks (ERWs) on $\mathbb{Z}$, each with a different memory parameter, can meet infinitely often, extending the work of Roy, Takei, and Tanemura. We also study the asymptotic behavior of their distance by providing an elementary and accessible proof of the classical Law of the Iterated Logarithm (LIL) for centered, continuous, self-similar Gaussian processes under a certain decay condition on the covariance kernel.