Selecting the last consecutive record in a record process

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 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 → ∞.