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 ψ-mixing conditions for Y. With probability one, n^-1log W _n(D) has the same limit as -n^-1logQ(B(X₁ⁿ, D)), where Q(B(X₁ ⁿ, D)) is the Q-measure of the D-ball around (X₁ ,...,X_n), with respect to a given distortion measure. Large deviations techniques are used to get the convergence of -n^-1 log Q(B(X₁ⁿ, D)). First, a sequence of functions R_n in terms of the marginal distributions of X₁ⁿ and Y₁ⁿ 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^-1logQ(B(X₁ⁿ , D)) is shown to converge to R(P, Q, D) with probability one