Three-Dimensional Moran Walk with Resets

Abstract

In this current paper, we propose to study a three-dimensional Moran model (Xn(1),Xn(2),Xn(3)), where each random walk (Xn(i))∈{1,2,3} increases by one unit or is reset to zero at each unit of time. We analyze the joint law of its final altitude Xn=max(Xn(1),Xn(2),Xn(3)) via the moment generating tools. Furthermore, we show that the limit distribution of each random walk follows a shifted geometric distribution with parameter 1−qi, and we analyze the maximum of these three walks, also giving explicit expressions for the mean and variance.