Stationary Probability Distributions for Linear Time-Invariant Systems

Abstract

Existence and properties of stationary probability distributions for the output vector of linear time-invariant systems perturbed by white noise are examined. It is shown that a necessary and sufficient condition for existence of such probability distribution is unobservability of the noise controlled unstable modes of the system. In particular, there exists a Gaussian stationary distribution for the output process under the above condition. This is the unique stationary probability distribution if and only if, in addition to the previous condition, all modes corresponding to zero or an imaginary characteristic value are unobservable. Convergence in distribution of the output process is examined, and equivalence of a given system and its dual (transposed system) with respect to existence of stationary probability distributions for their output processes is demonstrated.