  Reservoir computing systems are constructed using a driven dynamical system
in which external inputs can alter the evolving states of a system. These
paradigms are used in information processing, machine learning, and
computation. A fundamental question that needs to be addressed in this
framework is the statistical relationship between the input and the system
states. This paper provides conditions that guarantee the existence and
uniqueness of asymptotically invariant measures for driven systems and shows
that their dependence on the input process is continuous when the set of input
and output processes are endowed with the Wasserstein distance. The main tool
in these developments is the characterization of those invariant measures as
fixed points of naturally defined Foias operators that appear in this context
and which have been profusely studied in the paper. Those fixed points are
obtained by imposing a newly introduced stochastic state contractivity on the
driven system that is readily verifiable in examples. Stochastic state
contractivity can be satisfied by systems that are not state-contractive, which
is a need typically evoked to guarantee the echo state property in reservoir
computing. As a result, it may actually be satisfied even if the echo state
property is not present.
