Absolute continuity and mutual information for Gaussian mixtures

Abstract

Absolute continuity, process representations, and the Shannon information are considered for problems involving a Gaussian mixture process (Nt ),tin [0,1] a.e. dP(ω)dt, where (Gt ) is a Gaussian process and Ais a positive random variable independent of (Gt ). Let (Yt )tin [0, 1], be a second process with υYand υNthe measures induced on and μYand μNthe measures induced on L2[0, 1] (when (Yt) has paths a.s. in L2[0, 1] The Cramer–Hida spectral representation and an extension of Girsanov's theorem are used to obtain results on absolute continuity 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 (Nt ) as additive noise. Capacity is obtained for the no-feedback channel, subject to an average-energy type of constraint