POISSON APPROXIMATIONS FOR RUNS AND PATTERNS OF RARE EVENTS

Consider a sequence of Bernoulli trials with success probability p, and let N(n,k) denote the number of success runs of length k greater-than-or-equal-to 2 among the first n trials. The Stein-Chen method is employed to obtain a total variation upper bound for the rate of convergence of N(n,k) to a Poisson random variable under the standard condition np(k) --> lambda. 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(p(k-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.