### 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) |
---|---|

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 2009 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

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

**A study of counts of bernoulli strings via conditional poisson processes.** / Huffer, Fred W.; Sethuraman, Jayaram; Sethuraman, Sunder.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 137, no. 6, pp. 2125-2134. https://doi.org/10.1090/S0002-9939-08-09793-1

}

TY - JOUR

T1 - A study of counts of bernoulli strings via conditional poisson processes

AU - Huffer, Fred W.

AU - Sethuraman, Jayaram

AU - Sethuraman, Sunder

PY - 2009/6

Y1 - 2009/6

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=76449115113&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=76449115113&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-08-09793-1

DO - 10.1090/S0002-9939-08-09793-1

M3 - Article

AN - SCOPUS:76449115113

VL - 137

SP - 2125

EP - 2134

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 6

ER -