Poisson approximations for runs and patterns of rare events

Consider a sequence of Bernoulli trials with success probability p, and let Nn,k denote the number of success runs of length among the first n trials. The Stein.Chen method is employed to obtain a total variation upper bound for the rate of convergence of Nn,k to a Poisson random variable under the standard condition npk... This bound is of the same order, O(p), as the best known for the case k = 1, i.e. for the classical binomial-Poisson approximation. Analogous results are obtained for occurrences of word patterns, where, depending on the nature of the word, the corresponding rate is at most O(pk.m) for some m = 0, 2, ···, k . 1. The technique is adapted for use with two-state Markov chains. Applications to reliability systems and tests for randomness are discussed.