One-Dimensional Infinite Memory Imitation Models with Noise

Abstract

In this paper we study stochastic process indexed by Z constructed from certain transition kernels depending on the whole past. These kernels prescribe that, at any time, the current state is selected by looking only at a previous random instant. We characterize uniqueness in terms of simple concepts concerning families of stochastic matrices, generalizing the results previously obtained in De Santis and Piccioni (J. Stat. Phys., 150(6):1017-1029, 2013).