### Abstract

Suppose that I_{1}, I_{2}, ⋯ is a sequence of independent Bernoulli random variables with E(I_{n}) = λ/(λ + n - 1), n = 1, 2, ⋯. If λ is a positive integer k, {I_{n}}_{n}≥1 can be interpreted as a k-record process of a sequence of independent and identically distributed random variables with a common continuous distribution. When I_{n-1}I _{n} = 1, we say that a consecutive k-record occurs at time n. It is known that the total number of consecutive k-records is Poisson distributed with mean k. In fact, for general λ > 0, ∑^{∞} _{n=2} I_{n-1}I_{n} is Poisson distributed with mean λ. In this paper, we want to find an optimal stopping time τλ which maximizes the probability of stopping at the last n such that I _{n-1}I_{n} = 1. We prove that τλ is of threshold type, i.e. there exists a τλ ε ℕ such that τλ = min{n

Original language | English |
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Pages (from-to) | 739-760 |

Number of pages | 22 |

Journal | Advances in Applied Probability |

Volume | 42 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2010 Sep 1 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Applied Mathematics

### Cite this

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*Advances in Applied Probability*, vol. 42, no. 3, pp. 739-760. https://doi.org/10.1239/aap/1282924061

**Selecting the last consecutive record in a record process.** / Hsiau, Shoou Ren.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Selecting the last consecutive record in a record process

AU - Hsiau, Shoou Ren

PY - 2010/9/1

Y1 - 2010/9/1

N2 - Suppose that I1, I2, ⋯ is a sequence of independent Bernoulli random variables with E(In) = λ/(λ + n - 1), n = 1, 2, ⋯. If λ is a positive integer k, {In}n≥1 can be interpreted as a k-record process of a sequence of independent and identically distributed random variables with a common continuous distribution. When In-1I n = 1, we say that a consecutive k-record occurs at time n. It is known that the total number of consecutive k-records is Poisson distributed with mean k. In fact, for general λ > 0, ∑∞ n=2 In-1In is Poisson distributed with mean λ. In this paper, we want to find an optimal stopping time τλ which maximizes the probability of stopping at the last n such that I n-1In = 1. We prove that τλ is of threshold type, i.e. there exists a τλ ε ℕ such that τλ = min{n

AB - Suppose that I1, I2, ⋯ is a sequence of independent Bernoulli random variables with E(In) = λ/(λ + n - 1), n = 1, 2, ⋯. If λ is a positive integer k, {In}n≥1 can be interpreted as a k-record process of a sequence of independent and identically distributed random variables with a common continuous distribution. When In-1I n = 1, we say that a consecutive k-record occurs at time n. It is known that the total number of consecutive k-records is Poisson distributed with mean k. In fact, for general λ > 0, ∑∞ n=2 In-1In is Poisson distributed with mean λ. In this paper, we want to find an optimal stopping time τλ which maximizes the probability of stopping at the last n such that I n-1In = 1. We prove that τλ is of threshold type, i.e. there exists a τλ ε ℕ such that τλ = min{n

UR - http://www.scopus.com/inward/record.url?scp=78650951871&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78650951871&partnerID=8YFLogxK

U2 - 10.1239/aap/1282924061

DO - 10.1239/aap/1282924061

M3 - Article

AN - SCOPUS:78650951871

VL - 42

SP - 739

EP - 760

JO - Advances in Applied Probability

JF - Advances in Applied Probability

SN - 0001-8678

IS - 3

ER -