Selecting the last consecutive record in a record process

Suppose that I₁, I₂, ⋯ 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-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 I_n-1I_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-1I_n = 1. We prove that τλ is of threshold type, i.e. there exists a τλ ε ℕ such that τλ = min{n