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

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3 Author(s)

Let \xi = {\xi(t), 0 \leq t \leq T} be a process with covariance function K(s,t) and E \int_0^T \xi^2(t) dt < \infty . It is proved that for every \varepsilon > 0 the \varepsilon -entropy H_{\varepsilon }(\xi) satisfies begin{equation} H_{varepsilon}(xi_g) - mathcal{H}_{xi_g} (xi) leq H_{varepsilon}(xi) leq H_{varepsilon}(xi_g) end{equation} where \xi_g is a Gaussian process with the covarianee K(s,t) and mathcal{H}_{\xi_g}(\xi) is the entropy of the measure induced by \xi (in function space) with respect to that induced by \xi_g . It is also shown that if mathcal{H}_{\xi_g}(\xi) < \infty then, as \varepsilon \rightarrow 0 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 g = { g(t); 0 \leq t \leq T } such that mathcal{H}_g(\xi) < \infty , then the ratio between H_{\varepsilon }(\xi) and H_{\varepsilon }(g) goes to one as \varepsilon 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 mathcal_{\xi_g}(\xi) = \infty are discussed, and asymptotic bounds on H_{\varepsilon }(\xi) , expressed in terms of H_{\varepsilon }(\xi_g) , are derived.

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IEEE Transactions on Information Theory  (Volume:20 ,  Issue: 4 )