  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).
