Letxi = {xi(t), 0 leq t leq T}be a process with covariance functionK(s,t)andE int_0^T xi^2(t) dt < infty. It is proved that for everyvarepsilon > 0thevarepsilon-entropyH_{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} wherexi_gis a Gaussian process with the covarianeeK(s,t)andmathcal{H}_{xi_g}(xi)is the entropy of the measure induced byxi(in function space) with respect to that induced byxi_g. It is also shown that ifmathcal{H}_{xi_g}(xi) < inftythen, asvarepsilon rightarrow 0begin{equation} H_{varepsilon}(xi) = H_{varepsilon}(xi_g) - mathcal{H}_{xi_g}(xi) + o(1). end{equation} Furthermore, ff there exists a Gaussian processg = { g(t); 0 leq t leq T }such thatmathcal{H}_g(xi) < infty, then the ratio betweenH_{varepsilon}(xi)andH_{varepsilon}(g)goes to one asvarepsilongoes to zero. Similar results are given for the rate-distortion function, and some particular examples are worked out in detail. Some cases for whichmathcal_{xi_g}(xi) = inftyare discussed, and asymptotic bounds onH_{varepsilon}(xi), expressed in terms ofH_{varepsilon}(xi_g), are derived.

### Published in:

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

Jul 1974