One-dimensional infinite memory imitation models with noise

TL;DR

This paper investigates one-dimensional infinite memory stochastic models where the current state depends on a randomly chosen past instant, providing new criteria for uniqueness based on stochastic matrix properties.

Contribution

It generalizes previous results by characterizing uniqueness in infinite memory models using simple stochastic matrix concepts.

Findings

Established criteria for uniqueness in models with infinite memory.

Extended previous work to more general transition kernels.

Connected model behavior to properties of stochastic matrices.

Abstract

In this paper we study stochastic process indexed by $\mathbb {Z}$ 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).