Representation and smoothing of non-Gaussian Markov chains: a Kronecker-algebra based approach

In the present paper the class of finite-states, non- Gaussian, Markov-chains over a finite interval are considered. Under the hypothesis of complete knowledge of the process- statistics, and a nonsingularity assumption, the following results are proven: first by augmenting the process with all its Kronecker powers up to a certain degree (depending of the number of states) the augmented process can be stochastically realized by an ordinary stochastic recursive equation. Second, by supposing the process is partially and noisy observed by a linear equation, a smoothing algorithm is derived giving a smoothing estimate of the process which is optimal in a class of observations polynomial.