Abstract
This paper addresses the following problem: Given a possibly non-stationary second-order or Gaussian stochastic process over a certain time interval, find all state-space realizations that produce the given process as their output. A constructive solution to this problem is described. In particular, the degrees of freedom left in this problem are explicitly displayed in terms of additional random variables and processes injected into the state-space description. Special attention is paid to output-induced, or internal, state-space realizations, that is, those for which the state vector is completely determined by the given process. The development is self-contained, the approach and proofs are fairly elementary, and explicit formulas are given for the various realizations.
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