### Abstract

A sequence of random variables, each taking values 0 or 1, is called a Bernoulli sequence. We say that a string of length d occurs in a Bernoulli sequence if a success is followed by exactly (d - 1) failures before the next success. The counts of such d-strings are of interest, and in specific independent Bernoulli sequences are known to correspond to asymptotic d-cycle counts in random permutations. In this paper, we give a new framework, in terms of conditional Poisson processes, which allows for a quick characterization of the joint distribution of the counts of all d-strings, in a general class of Bernoulli sequences, as certain mixtures of the product of Poisson measures. In particular, this general class includes all Bernoulli sequences considered in the literature, as well as a host of new sequences.

Original language | English (US) |
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Pages (from-to) | 2125-2134 |

Number of pages | 10 |

Journal | Proceedings of the American Mathematical Society |

Volume | 137 |

Issue number | 6 |

DOIs | |

State | Published - Jun 1 2009 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Proceedings of the American Mathematical Society*,

*137*(6), 2125-2134. https://doi.org/10.1090/S0002-9939-08-09793-1