An urn contains balls of different colours, which are randomly drawn one at a time. After each draw the number of balls in the urn with the same colour as the ball last drawn is changed. Special cases are sampling with and without replacement, and Pólya sampling. The drawing stops when a given number of colours has been drawn a given number of times. The number of times the different colours have been drawn is studied in this paper by imbedding the urn scheme in birth processes. Both exact and asymptotic results are obtained. In particular, waiting times for sampling with and without replacement and for Pólya sampling are considered.

Strong mixing is a condition which is often assumed to prove limit theorems for strictly stationary processes. Leadbetter's condition D(un) is used to prove limit theorems for maxima of stationary processes.

A sufficient condition for strong mixing to hold is given for the case where the process satisfies a pth-order Markov property. This condition can be easy to check for when p is small. This point is illustrated by two examples of first-order autoregressive processes.

The condition D(un) is shown to hold for any stationary Markov process.