On the -entropy and the rate-distortion function of certain non-Gaussian processes

Let be a process with covariance function and . It is proved that for every the -entropy satisfies begin{equation} H_{varepsilon}(xi_g) - mathcal{H}_{xi_g} (xi) leq H_{varepsilon}(xi) leq H_{varepsilon}(xi_g) end{equation} where is a Gaussian process with the covarianee and is the entropy of the measure induced by (in function space) with respect to that induced by . It is also shown that if then, as begin{equation} H_{varepsilon}(xi) = H_{varepsilon}(xi_g) - mathcal{H}_{xi_g}(xi) + o(1). end{equation} Furthermore, ff there exists a Gaussian process such that , then the ratio between and goes to one as goes to zero. Similar results are given for the rate-distortion function, and some particular examples are worked out in detail. Some cases for which are discussed, and asymptotic bounds on , expressed in terms of , are derived.