Chryssaphinou, O.; Papastavridis, S. A limit theorem for the number of non-overlapping occurrences of a pattern in a sequence of independent trials. (English) Zbl 0643.60020 J. Appl. Probab. 25, No. 2, 428-431 (1988). Y\({}_ 1,Y_ 2,..\). is an infinite sequence of independent, identically distributed random variables, with possible values \(w_ 1,...,w_ q\), these being q distinct symbols in an alphabet, \(q\geq 2\). \(P(Y_ i=w_ j)\) is denoted by \(p_ j\). A pattern is a sequence of letters from the alphabet. For each positive integer n, suppose a pattern \(A_ n\) of length \(k_ n\) is specified, where \(k_ n\) approaches infinity as n approaches infinity. Let \(P(A_ n)\) denote the probability that pattern \(A_ n\) occurs at specified positions. Let \(T_ n\) denote the number of nonoverlapping occurrences of \(A_ n\) among the first n Y’s. Define \(q_{i,n}\) as the conditional probability that \(A_ n\) occurs with an overlap i after a given \(A_ n\). Define \(u_ n\) as \((\sum_{i}q_{i,n})/P(A_ n)\). The following theorem is proved. If \(n/u_ n\) approaches L as n increases, with L positive, then the distribution of \(T_ n\) approaches a Poisson distribution with parameter L. Reviewer: L.Weiss Cited in 6 Documents MSC: 60F05 Central limit and other weak theorems Keywords:symbols in an alphabet; number of nonoverlapping occurrences; Poisson distribution PDFBibTeX XMLCite \textit{O. Chryssaphinou} and \textit{S. Papastavridis}, J. Appl. Probab. 25, No. 2, 428--431 (1988; Zbl 0643.60020) Full Text: DOI