Godbole, Anant P. Poisson approximations for runs and patterns of rare events. (English) Zbl 0751.60018 Adv. Appl. Probab. 23, No. 4, 851-865 (1991). 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) Cited in 1 ReviewCited in 18 Documents 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) Keywords:Poisson approximations; word patterns; sequence of Bernoulli trials; number of non-overlapping success runs; two-state Markov chains; reliability systems PDFBibTeX XMLCite \textit{A. P. Godbole}, Adv. Appl. Probab. 23, No. 4, 851--865 (1991; Zbl 0751.60018) Full Text: DOI