Page 28 - Contributed Paper Session (CPS) - Volume 4

CPS2107 Junfan Tao et al.

Our purpose here is to study the asymptotic behavior of and

.
The contributions of the present paper are as follows.
First, for the stationary AR(1), we prove the joint asymptotic normality of
the sequential estimators for and the stopping times τ1c in (5) and τ2c in
(10). Especially, we nd that τ1c has the asymptotic variance strictly greater than
τ2c.
Second, we introduce the following new methodology in sequential
analysis. We represent random quantities of concern in terms of stochastic
processes in D[0,∞) and apply functional central limit theorems in D[0,∞)
including Theorem 2 which is an extension of Theorem 19.1 in Billingsley(1999,
p.197). Together with the limits of the stopping times, we derive the
asymptotic properties of the sequential statistics. Skorohod representation
theorem (Billingsley (1999, p.70)) makes it possible to evaluate the continuous-
time stochastic processes represented by Brownian motions at the limits of the
stopping times.

2. Methodology
According to Lai and Siegmund (1983), τ1c/c → 1 − β almost surely. We
2
2
show the same result holds for τ2c/c.
Theorem 1. Let x0 be an arbitrary L random variables and independent of
2
. Then, τ1c defined in (5) and τ2c in (10) satisfy:
(11)

(12)

2
The sequential estimates of β and σ have strong consistency;
2
lim ̂ = lim ̂ = . and lim 2 = . (13)
→∞ 1 →∞ 2 →∞ 2
Theorem 1 gives the almost sure convergence of the stopping times τ1c
and τ2c. Now we provide some asymptotics with respect to the sequential
statistics. We consider applying the theory of convergence of random
elements in D[0,∞); D[0,∞) is the set of the right continuous functions on
[0,∞) with left limits. In a sequential sampling scheme, the space D[0,∞) is
natural to characterize the limiting behavior of sequential statistics, since we
consider stopping times with unbounded range of integers.

The following theorem and Skorohod’s representation theorem
(Billingsley(1999, p.70).) allows us to derive the joint asymptotic normality of

2
2 limc→∞ τ1c/c = limc→∞ τ2c/c = σ /γ(0) also holds for any stationary p-th order autoregressive process
with covariance function γ(m).
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