Elephant random walks with graph based shared memory: First and second order asymptotics

We consider a generalization of the so-called elephant random walk by introducing multiple elephants moving along the integer line, $\mathbb{Z}$. When taking a new step, each elephant considers not only its own previous steps but also the past steps of other elephants. The dynamics of "who follows whom" are governed by a directed graph, where each vertex represents an elephant, and the edges indicate that an elephant will consider the past steps of its in-neighbour elephants when deciding its next move. In other words, this model involves a collection of reinforced random walks evolving through graph-based interactions. We briefly investigate the first- and second-order asymptotic behaviour of the joint walks and establish connections with other network-based reinforced stochastic processes studied in the literature. We show that the joint walk can be expressed as a stochastic approximation scheme. In certain regimes, we employ tools from stochastic approximation theory to derive the asymptotic properties of the joint walks. Additionally, in a specific regime, we use better techniques to establish a strong invariance principle and a central limit theorem with improved rates compared to existing results in the stochastic approximation literature. These techniques can also be used to strengthen equivalent results in stochastic approximation theory. As a byproduct, we establish a strong invariance principle for the simple elephant random walk with significantly improved rates.