State-Space Systems as Dynamic Generative Models
Juan-Pablo Ortega Florian Rossmannek
A probabilistic framework to study the dependence structure induced by
deterministic discrete-time state-space systems between input and output
processes is introduced. General sufficient conditions are formulated under
which output processes exist and are unique once an input process has been
fixed, a property that in the deterministic state-space literature is known as
the echo state property. When those conditions are satisfied, the given
state-space system becomes a generative model for probabilistic dependences
between two sequence spaces. Moreover, those conditions guarantee that the
output depends continuously on the input when using the Wasserstein metric. The
output processes whose existence is proved are shown to be causal in a specific
sense and to generalize those studied in purely deterministic situations. The
results in this paper constitute a significant stochastic generalization of
sufficient conditions for the deterministic echo state property to hold, in the
sense that the stochastic echo state property can be satisfied under
contractivity conditions that are strictly weaker than those in deterministic
situations. This means that state-space systems can induce a purely
probabilistic dependence structure between input and output sequence spaces
even when there is no functional relation between those two spaces.