The Fisher-Darmois-Koopman-Pitman theorem for random processes

The Fisher-Darmois-Koopman-Pitman theorem says that for smooth nowhere vanishing probability densities, a finite dimensional sufficient statistic exists if and only if the densities are from an exponential family. This is a classical result for parameter estimation. It has also been proved for a limited class of random processes. In this paper, it is proved for a very wide class of random processes.