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**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) |

### Keywords:

binary trials; exchangeable trials; independent trials; runs; strings; waiting time; urn models; records; nonparametric tests of randomness
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\textit{F. S. Makri} and \textit{Z. M. Psillakis}, J. Stat. Plann. Inference 142, No. 4, 908--924 (2012; Zbl 1232.62071)

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