Selecting the last consecutive record in a record process

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−1In = 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 In−1In = 1. We prove that τλ is of threshold type, i.e. there exists a tλ ∈ ℕ such that τλ = min{n | n ≥ tλ, In−1In = 1}. We show that tλ is increasing in λ and derive an explicit expression for tλ. We also compute the maximum probability Qλ of stopping at the last consecutive record and study the asymptotic behavior of Qλ as λ → ∞.