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Absolute continuity and mutual information for Gaussian mixtures. (English)
Stochastics Stochastics Rep. 39, No.2-3, 139-157 (1992).
Summary: Absolute continuity, process representations, and the Shannon information are considered for problems involving a Gaussian mixture process $(N\sb t)$, $t$ in [0,1], $N(ω,t)=A(ω)G(ω,t)$, where $(G\sb t)$ is a Gaussian process and $A$ is a positive random variable independent of $(G\sb t)$. Let $(Y\sb t)$, $t$ in [0,1], be a second process with $ν\sb Y$ and $ν\sb N$ the measures induced on ${\bbfR}\sp{[0,1]}$ and $μ\sb Y$ and $μ\sb N$ the measures induced on $L\sb 2[0,1]$ (when $(Y\sb t)$ has paths a.s. in ${\cal L}\sb 2[0,1]$). The Cramér-Hida spectral representation and an extension of Girsanov’s theorem are used to obtain results on absolute continuity ($ν\sb Y\llν\sb N$ and $μ\sb Y\ll μ\sb N$) 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 $(N\sb t)$ as additive noise. Capacity is obtained for the no-feedback channel, subject to an average-energy type of constraint.