The elephant random walk in the triangular array setting

Gut and Stadmüller (2021, 2022) initiated the study of the elephant random walk with limited memory. Aguech and El Machkouri (2024) published a paper in which they discuss an extension of the results by Gut and Stadtmüller (2022) for an 'increasing memory' version of the elephant random walk without stops. Here we present a formal definition of the process that was hinted at by Gut and Stadtmüller. This definition is based on the triangular array setting. We give a positive answer to the open problem in Gut and Stadtmüller (2022) for the elephant random walk, possibly with stops. We also obtain the central limit theorem for the supercritical case of this model.