Sufficient sequences and state-space models for random processes

Abstract

Let ( U_{1}, U_{2} , ...) be a sequence of observed random variables whose probability distributions are described by a parameterized family of density functions \{p_{k}(u_{1}, ..., u_{k}; \theta)\} . If there exists a sequence of sufficient statistics for \theta(T_{1}(U_{1}) , T_{2}(U_{1}, U_{2}) , ...), and if a realizability assumption holds, then there is a finite-dimensional state-space model whose output process agrees with ( U_{1}, U_{2} , ...) in distribution.