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On occurrences of \(F-S\) strings in linearly and circularly ordered binary sequences. (English) Zbl 1191.60018

Summary: Consider a sequence of exchangeable or independent binary trials ordered on a line or on a circle. The statistics denoting the number of times an \(F-S\) string of length (at least) \(k_{1} + k_{2}\), that is, (at least) \(k_{1}\) failures followed by (at least) \(k_{2}\) successes in \(n\) such trials, are studied. The associated waiting time for the \(r\)th occurrence of an \(F-S\) string of length (at least) \(k_{1} + k_{2}\) in linearly ordered trials is also examined. Exact formulae, lower/upper bounds and approximations are derived for their distributions. Mean values and variances of the number of occurrences of \(F-S\) strings are given in exact formulae too. Particular exchangeable and independent sequences of binary random variables, used in applied research, combined with numerical examples clarify further the theoretical results.

MSC:

60E05 Probability distributions: general theory
62E15 Exact distribution theory in statistics
60C05 Combinatorial probability
60G09 Exchangeability for stochastic processes
60F05 Central limit and other weak theorems
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