1 results
Poisson approximations for runs and patterns of rare events
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- Journal:
- Advances in Applied Probability / Volume 23 / Issue 4 / December 1991
- Published online by Cambridge University Press:
- 01 July 2016, pp. 851-865
- Print publication:
- December 1991
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- Article
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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→λ. 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.
among the first 