Innovations informational equivalence for observations with non-Gaussian noise

This paper considers the problem of innovations informational equivalence for the observations with non-Gaussian noise. It is assumed that the additive noise is a non-Gaussian martingale which is dependent on a finite state Markov chain. Under the assumptions of stochastic independence between the signal and the noise, and of a weighted square integrability on the signal, it is shown that the non-Gaussian innovations process, i.e., a non-Gaussian martingale adapted to the observation, is informationally equivalent to the observation. It is also shown that no Gaussian martingale (Brownian motion process) is informationally equivalent to the observation.