State space models with finite dimensional dependence (Q2784954)

The authors consider a model in which the observable (multivariate) process \(Y_t\) depends on a stationary multivariate state process \(\zeta_t\) and the conditional distribution of \(\zeta_t\) for fixed \(\zeta_{t-1}\), \(\zeta_{t-2}\), \dots, \(y_{t-1}\), \(y_{t-2}\), \dots is of the form NEWLINE\[NEWLINE\pi(\zeta_t \;|\;\zeta_{t-1})=\pi(\zeta_t)\sum_{j=1}^J a_j(\zeta_t)b_j(\zeta_{t-1}).NEWLINE\]NEWLINE Such processes are called FDD-processes. For FDD models the authors derive explicit formulas for filtering, prediction and smoothing of the observable process. It is shown that a stationary Markov process with continuous transition function has finite-dimensional predictor space iff it possesses the FDD-property. A characterization of FDD-processes in terms of nonlinear canonical correlations is also considered.