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Poisson approximations for runs and patterns of rare events. (English) Zbl 0751.60018

A sequence of Bernoulli trials is considered. The number of overlapping occurrences \((M_{n,k})\) of a fixed word of length \(k\) is considered. The alphabet is of size \(m_ n\) and \(n\) letters are drawn. Each successive letter is chosen independently with probability \(p_ n=1/m_ n\). The other viewpoint supposes, that the \(k\) letters of the word in question each have a probability \(p_ n\) of being selected. Let \(N_{n,k}\) be the number of non-overlapping success runs of length \(k\) among the first \(n\) trials. It is proved \(M_{n,k}\Rightarrow Po(\lambda)\) if \(np^ k_ n\to \lambda\) (\(M_{n,k}\) tends to the random variable degenerate at 0 if \(nf(p_ n)\to \lambda\), where \(p^ k_ n=o(f(p_ n))\)). The convergence \(N_{n,k}\) to \(Po(\lambda)\) is also studied. The analogs of these results for two-state Markov chains are also proved. Applications: standard runs tests, reliability systems.
Reviewer: P.Froněk (Praha)

MSC:

60F05 Central limit and other weak theorems
60K10 Applications of renewal theory (reliability, demand theory, etc.)
62G10 Nonparametric hypothesis testing
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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