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

-Let C = {5(t), 0 5 t I T} be a process with covariance function K(s,t ) and E jf t'(t) dt < co. It is proved that for every E > 0 the E-entropy H,(T) satisfies where & is a Gaussian process with the covarhmce K&t) and &‘;,(T) is the entropy of the measure induced by c (in function space) with respect to that induced by &,. It is also shown that if ze,(T) < co, then, as E + 0 Furthermore, if there exists a Gaussian process g = {g(t); 0 5 t 5 T} such that y%(T) < co, then the ratio between H,(5) and H,(g) goes to one as E 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 A$,(@ = 00 are discussed, and asymptotic bounds on H,(T), expressed in terms of H&&J, are derived.