  The capability of recurrent neural networks to approximate trajectories of a
random dynamical system, with random inputs, on non-compact domains, and over
an indefinite or infinite time horizon is considered. The main result states
that certain random trajectories over an infinite time horizon may be
approximated to any desired accuracy, uniformly in time, by a certain class of
deep recurrent neural networks, with simple feedback structures. The
formulation here contrasts with related literature on this topic, much of which
is restricted to compact state spaces and finite time intervals. The model
conditions required here are natural, mild, and easy to test, and the proof is
very simple.
