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-n(D) between X and Y, allowing distortion, is established in the presence of one-sided psi -mixing conditions for Y. With probability one, n(-1) log W-n(D) has the same limits -n(-1) log Q(B(X-1(n), D)), where Q(B(X-1(n), D)) is the Q-measure of the D-ball around (X-1,..., X-n), with respect to a given distortion measure. Large deviations techniques are used to get the convergence of -n(-1) log Q(B(X-1(n), D)). First, a sequence of functions R-n in terms of the marginal distributions of X-1(n) and Y-1(n) as well as D are constructed and demonstrated to converge to a function R(P, Q, D). The functions R-n and R(P, Q, D) are different from rate distortion functions. Then -n(-1) log Q(B(X-1(n), D)) is shown to converge to R(P, Q, D) with probability one.