Source: Adv. in Appl. Probab. Volume 42, Number 3
(2010), 739-760.
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
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
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 → ∞.
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