Sufficient sequences and state-space models for random processes

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.