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Selecting the last consecutive record in a record process

Part of: Stochastic processes Sequential methods Special processes

Published online by Cambridge University Press:  01 July 2016

Shoou-Ren Hsiau
Shoou-Ren Hsiau*
Affiliation:
National Changhua University of Education
*
∗ Postal address: Department of Mathematics, National Changhua University of Education, No. 1, Jin-De Road, Changhua 500, Taiwan, R.O.C. Email address: srhsiau@cc.ncue.edu.tw
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Abstract

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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 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 t λ ∈ ℕ such that τλ = min{n | n ≥ t λ, I n−1 I n = 1}. We show that t λ is increasing in λ and derive an explicit expression for t λ. We also compute the maximum probability Q λ of stopping at the last consecutive record and study the asymptotic behavior of Q λ as λ → ∞.

Keywords

Optimal stoppingthreshold typeconsecutive recordmonotone stopping rulerecord process

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 62L15: Optimal stopping 60K99: None of the above, but in this section

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 3 , September 2010 , pp. 739 - 760
Copyright
Copyright © Applied Probability Trust 2010 

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