Poisson approximations for runs and patterns of rare events

Consider a sequence of Bernoulli trials with success probabilityp, and letNn,kdenote the number of success runs of length among the firstntrials. The Stein–Chen method is employed to obtain a total variation upper bound for the rate of convergence ofNn,kto a Poisson random variable under the standard conditionnpk→λ. This bound is of the same order,O(p), as the best known for the casek =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 mostO(pk–m) for somem= 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.