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A limit theorem for the number of non-overlapping occurrences of a pattern in a sequence of independent trials. (English) Zbl 0643.60020

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

MSC:

60F05 Central limit and other weak theorems
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