Absolute Continuity and Mutual Information for Gaussian Mixtures.

摘要

Absolute continuity, process representations, and the Shannon information are considered for problems involving a Gaussian mixture process (N sub t), t in (0,1). N(omega,t)=A(omega)G(omega,t) a.e. dP(omega)dt, where (G sub t) is a Gaussian process and A is a positive random variable independent of (G sub t). Let (Y sub t), t in 0,1, be a second process with nu sub Y and nu sub N the measures induced on R0,1 and mu sub Y and mu sub N the measures induced on L20,1 (Y sub t) has paths a.s. in L20.1. The Cramer-Hida spectral representation and an extension of Girsanov's theorem are used to obtain results on absolute continuity (nu sub Y << nu sub N and mu sub Y << mu sub N) and likelihood ratio in terms of similar results involving a Gaussian mixture local martingale, for which representations are given. These results are then applied to obtain the Shannon mutual information for a communication channel with feedback having (N sub t) as additive noise. Capacity is obtained for the no-feedback channel, subject to an average-energy type of constraint. (jhd)