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**Joint distributions of counts of strings in finite Bernoulli sequences.**
*(English)*
Zbl 1314.60034

Summary: An infinite sequence \((Y_{1}, Y_{2},\dots )\) of independent Bernoulli random variables with \(P(Y_{i} = 1) = a / (a + b + i - 1)\), \(i = 1, 2,\dots \), where \(a > 0\) and \(b \geq 0\), will be called a \(\mathrm{Bern}(a, b)\) sequence. Consider the counts \(Z_{1}, Z_{2}, Z_{3},\dots \) of occurrences of patterns or strings of the form \(\{11\}, \{101\}, \{1001\},\dots\), respectively, in this sequence. The joint distribution of the counts \(Z_{1}, Z_{2},\dots \) in the infinite \(\mathrm{Bern}(a, b)\) sequence has been studied extensively. The counts from the initial finite sequence \((Y_{1}, Y_{2},\dots , Y_{n})\) have been studied by L. Holst [ibid. 44, No. 3, 824–830 (2007; Zbl 1132.60011); ibid. 45, No. 3, 901–906 (2008; Zbl 1151.60002)], who obtained the joint factorial moments for \(\mathrm{Bern}(a, 0)\) and the factorial moments of \(Z_{1}\), the count of the string \(\{1, 1\}\), for a general \(\mathrm{Bern}(a, b)\). We consider stopping the Bernoulli sequence at a random time and describe the joint distribution of counts, which extends Holst’s results. We show that the joint distribution of counts from a class of randomly stopped Bernoulli sequences possesses the mixture of independent Poissons property: there is a random vector conditioned on which the counts are independent Poissons. To obtain these results, we extend the conditional marked Poisson process technique introduced by F. W. Huffer et al. [Proc. Am. Math. Soc. 137, No. 6, 2125–2134 (2009; Zbl 1165.60305)]. Our results avoid previous combinatorial and induction methods which generally only yield factorial moments.

### Keywords:

conditional marked Poisson process; Bernoulli sequence; counts of strings; random permutation; cycles; flaws and failures
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\textit{F. W. Huffer} and \textit{J. Sethuraman}, J. Appl. Probab. 49, No. 3, 758--772 (2012; Zbl 1314.60034)

### References:

[1] | Arratia, R., Barbour, A. D. and Tavaré, S. (1992). Poisson process approximations for the Ewens sampling formula. Ann. Appl. Prob. 2, 519-535. · Zbl 0756.60006 |

[2] | Arratia, R., Barbour, A. D. and Tavaré, S. (2003). Logarithmic Combinatorial Structures: A Probabilistic Approach. European Mathematical Society, Zürich. · Zbl 1040.60001 |

[3] | Hahlin, L.-O. (1995). Double records. Res. Rep. 1995:12, Department of Mathematics, Uppsala University. |

[4] | Holst, L. (2007). Counts of failure strings in certain Bernoulli sequences. J. Appl. Prob. 44, 824-830. · Zbl 1132.60011 |

[5] | Holst, L. (2008a). A note on embedding certain Bernoulli sequences in marked Poisson processes. J. Appl. Prob. 45, 1181-1185. · Zbl 1154.60009 |

[6] | Holst, L. (2008b). The number of two consecutive successes in a Hoppe-Pólya urn. J. Appl. Prob. 45, 901-906. · Zbl 1151.60002 |

[7] | Holst, L. (2009). On consecutive records in certain Bernoulli sequences. J. Appl. Prob. 46, 1201-1208. · Zbl 1187.60006 |

[8] | Holst, L. (2011). A note on records in a random sequence. Ark. Mat. 49, 351-356. · Zbl 1254.60017 |

[9] | Huffer, F., Sethuraman, J. and Sethuraman, S. (2008). A study of counts of Bernoulli strings via conditional Poisson processes. Preprint. Available at http://arxiv.org/abs/0801.2115v1. · Zbl 1165.60305 |

[10] | Huffer, F., Sethuraman, J. and Sethuraman, S. (2009). A study of counts of Bernoulli strings via conditional Poisson processes. Proc. Amer. Math. Soc. 137, 2125-2134. · Zbl 1165.60305 |

[11] | Joffe, A., Marchand, É., Perron, F. and Popadiuk, P. (2004). On sums of products of Bernoulli variables and random permutations. J. Theoret. Prob. 17, 285-292. · Zbl 1054.60013 |

[12] | Kingman, J. F. C. (1993). Poisson Processes. Oxford University Press. · Zbl 0771.60001 |

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

[14] | Resnick, S. I. (1992). Adventures in Stochastic Processes. Birkhäuser, Boston, MA. · Zbl 0762.60002 |

[15] | Sethuraman, J. and Sethuraman, S. (2004). On counts of Bernoulli strings and connections to rank orders and random permutations. In A festschrift for Herman Rubin (IMS Lecture Notes Monogr. Ser. 45 ), Institute for Mathematical Statistics, Beachwood, OH, pp. 140-152. \endharvreferences · Zbl 1268.60011 |

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