The first-order asymptotic of waiting times with distortion between stationary processes

Let X and Y be two independent stationary processes on general metric spaces, with distributions P and Q, respectively. The first-order asymptotic of the waiting time W/sub n/(D) between X and Y, allowing distortion, is established in the presence of one-sided /spl psi/-mixing conditions for Y. With probability one, n/sup -1/log W/sub n/(D) has the same limit as -n/sup -1/logQ(B(X/sub 1//sup n/, D)), where Q(B(X/sub 1//sup n/, D)) is the Q-measure of the D-ball around (X/sub 1/,...,X/sub n/), with respect to a given distortion measure. Large deviations techniques are used to get the convergence of -n/sup -1/log Q(B(X/sub 1//sup n/, D)). First, a sequence of functions R/sub n/ in terms of the marginal distributions of X/sub 1//sup n/ and Y/sub 1//sup n/ as well as D are constructed and demonstrated to converge to a function R(P, Q, D). The functions R/sub n/ and R(P, Q, D) are different from rate distortion functions. Then -n/sup -1/logQ(B(X/sub 1//sup n/, D)) is shown to converge to R(P, Q, D) with probability one.