## Counting certain binary strings.(English)Zbl 1232.62071

Summary: Consider a sequence of exchangeable or independent binary (i.e., zero-one) random variables. Numbers of strings with a fixed number of ones between two subsequent zeros are studied under an overlapping enumeration scheme. The respective waiting times are examined as well. Exact probability functions are obtained by means of combinatorial analysis and via recursive schemes in the case of an exchangeable and of an independent sequence, respectively. Explicit formulae for the mean values and variances of the number of strings are provided for both types of the sequences. For a Bernoulli sequence the asymptotic normality of the numbers of strings is established too. Indicative exchangeable and independent sequences, combined with numerical examples, clarify further the theoretical results.

### MSC:

 62G10 Nonparametric hypothesis testing 60G09 Exchangeability for stochastic processes 62E15 Exact distribution theory in statistics 65C60 Computational problems in statistics (MSC2010)
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### References:

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