  The universal approximation properties with respect to $L ^p $-type criteria
of three important families of reservoir computers with stochastic
discrete-time semi-infinite inputs is shown. First, it is proved that linear
reservoir systems with either polynomial or neural network readout maps are
universal. More importantly, it is proved that the same property holds for two
families with linear readouts, namely, trigonometric state-affine systems and
echo state networks, which are the most widely used reservoir systems in
applications. The linearity in the readouts is a key feature in supervised
machine learning applications. It guarantees that these systems can be used in
high-dimensional situations and in the presence of large datasets. The $L ^p $
criteria used in this paper allow the formulation of universality results that
do not necessarily impose almost sure uniform boundedness in the inputs or the
fading memory property in the filter that needs to be approximated.
