Selecting the last consecutive record in a record process

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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

Original languageEnglish
Pages (from-to)739-760
Number of pages22
JournalAdvances in Applied Probability
Issue number3
Publication statusPublished - 2010 Sep 1

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Applied Mathematics

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