Summary: Almost-sure convergence of \((1/k) \log W_k(x,y)\) to entropy for weak Bernoulli processes is proved, where \(W_k (x,y)\) is the waiting time until an initial segment of length \(k\) of a sample path \(x\) is seen in an independently chosen sample path \(y\). Analogous almost-sure results are obtained in the approximate match case for very weak Bernoulli processes. The weak Bernoulli proof uses recent results obtained by the authors about the estimation of joint distributions [Ann. Probab. 22, No. 2, 960- 977 (1994; Zbl 0806.28014)], while the very weak Bernoulli result utilizes a new characterization of such processes in terms of a blowing- up property.