##
**A study of counts of Bernoulli strings via conditional Poisson processes.**
*(English)*
Zbl 1165.60305

Summary: 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.

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.

### Keywords:

Bernoulli; cycles; strings; spacings; nonhomogeneous; Poisson processes; random permutations
PDF
BibTeX
XML
Cite

\textit{F. W. Huffer} et al., Proc. Am. Math. Soc. 137, No. 6, 2125--2134 (2009; Zbl 1165.60305)

### References:

[1] | Richard Arratia, A. D. Barbour, and Simon Tavaré, Poisson process approximations for the Ewens sampling formula, Ann. Appl. Probab. 2 (1992), no. 3, 519 – 535. · Zbl 0756.60006 |

[2] | Richard Arratia, A. D. Barbour, and Simon Tavaré, Logarithmic combinatorial structures: a probabilistic approach, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2003. · Zbl 1040.60001 |

[3] | Richard Arratia and Simon Tavaré, The cycle structure of random permutations, Ann. Probab. 20 (1992), no. 3, 1567 – 1591. · Zbl 0759.60007 |

[4] | Hua-Huai Chern, Hsien-Kuei Hwang, and Yeong-Nan Yeh, Distribution of the number of consecutive records, Proceedings of the Ninth International Conference ”Random Structures and Algorithms” (Poznan, 1999), 2000, pp. 169 – 196. , https://doi.org/10.1002/1098-2418(200010/12)17:3/43.0.CO;2-K · Zbl 0969.60017 |

[5] | W. Feller, The fundamental limit theorems in probability, Bull. Amer. Math. Soc. 51 (1945), 800 – 832. · Zbl 0060.28702 |

[6] | J. K. Ghosh and R. V. Ramamoorthi, Bayesian nonparametrics, Springer Series in Statistics, Springer-Verlag, New York, 2003. · Zbl 1029.62004 |

[7] | Lars Holst, Counts of failure strings in certain Bernoulli sequences, J. Appl. Probab. 44 (2007), no. 3, 824 – 830. · Zbl 1132.60011 |

[8] | Anatole Joffe, Éric Marchand, François Perron, and Paul Popadiuk, On sums of products of Bernoulli variables and random permutations, J. Theoret. Probab. 17 (2004), no. 1, 285 – 292. · Zbl 1054.60013 |

[9] | Kolchin, V.F. (1971), A problem of the allocation of particles in cells and cycles of random permutations. Theory Probab. Appl. 16 74-90. · Zbl 0239.60014 |

[10] | Ramesh M. Korwar and Myles Hollander, Contributions to the theory of Dirichlet processes, Ann. Probability 1 (1973), 705 – 711. · Zbl 0264.60084 |

[11] | Tamás F. Móri, On the distribution of sums of overlapping products, Acta Sci. Math. (Szeged) 67 (2001), no. 3-4, 833 – 841. · Zbl 1001.60015 |

[12] | Sidney Resnick, Adventures in stochastic processes, Birkhäuser Boston, Inc., Boston, MA, 1992. · Zbl 0762.60002 |

[13] | Jayaram Sethuraman and Sunder Sethuraman, On counts of Bernoulli strings and connections to rank orders and random permutations, A festschrift for Herman Rubin, IMS Lecture Notes Monogr. Ser., vol. 45, Inst. Math. Statist., Beachwood, OH, 2004, pp. 140 – 152. · Zbl 1268.60011 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.