Almost sure waiting time results for weak and very weak Bernoulli processes

Asymptotic properties of the waiting time W_k(x, y) until an initial segment of length k of a sample path x of an ergodic finite-alphabet process is seen in an independently chosen sample path y were discussed in earlier papers by Wyner and Ziv (1985), by Noble and Wyner (see IEEE Trans. Inform. Theory, vol.IT-38, p.1551, 1993), and by Shields (see The Journal of Theoretical Probability, vol.6, p.499-519, 1993). These results are sharpened to obtain almost sure convergence in the weak Bernoulli case for exact matches and in the very weak Bernoulli case for approximate matches. The weak Bernoulli proof uses recent results obtained by the authors about estimation of joint distributions, while the very weak Bernoulli result utilizes a new characterization of such processes in terms of a blowing-up property