Mutual information rate, distortion, and quantization in metric spaces

Abstract

Several new properties as well as simplified proofs of known properties are developed for the mutual information rate between discrete-time random processes whose alphabets are Borel subsets of complete separable metric spaces. In particular, the asymptotic properties of quantizers for such spaces provide a fink with finite-alphabet processes and yield the ergodic decomposition of mutual information rate. This result is used to prove the equality of stationary and ergodic process distortion-rate functions with the usual distortion-rate function. An unusual definition of mutual information rate for continuous-alphabet processes is used, but it is shown to be operationally appropriate and more useful mathematically; it provides an intuitive link between continuous-alphabet and finite-alphabet processes, and it allows generalizations of some fundamental results of ergodic theory that are useful for information theory.