Wyner and Ziv (1989) studied the asymptotic properties of recurrence times of stationary ergodic processes, and applied the results to obtain optimal data compression schemes in information transmission. Since then many data compression algorithms based upon string matching of a sequence from an information source with a database have been proposed and studied. In this paper we consider Gaussian stationary processes representing an information source and a database, and study problems of string matching with distortion. We prove theorems concerning the asymptotic behavior of the probability of string matching with distortion and the waiting time for the string matching.

The main purpose of this paper is to give a method to evaluate the actual value of the filtering error which arises in the transmission of a signal process, using the optimal coding, over a Gaussian channel. In his earier papers ([4] and [7]), the author has shown a method to construct an optimal causal coding for which the filtering error is minimized and at the same time the mutual information is maximized.

Let be a stochastic process, where T is a finite interval. The ε-entropy is defined as the following quantity:

(1)