Page 30 - Contributed Paper Session (CPS) - Volume 4

CPS2107 Junfan Tao et al.
Using the above lemma, we obtain
2
Theorem 5. (Sequential Asymptotic Normality) Let x0 be an arbitrary L random
variable and independent of with . As c → ∞,
in the sense of D[0,∞),

, (20)
and

.(21)
The following corollary shows the asymptotic variance of τ1c is strictly greater
than that of τ2c.
Under the same assumption in Lemma 4. For the stopping time τ1c and τ2c de
ned in (5) and (10), as c ↑ ∞

(22)

. (23)
2
Using the following proposition, one can obtain the long-run variance ν in (17).
Proposition 6. For a strongly stationary AR(1) process {xn} de ned in (1) with the
covariance function γ(m) and is nite. Let ,
.
2
2m
4
When µ4 = 3σ , then γx 2(m) = 2γ(m) , γx 2(m) = β γx 2(0) and the long-run variance
in (17)
. (24)

4. Discussion and Conclusion
Now, we provide a simulation study to examine our main results in Lemma
4 and Theorem 5. The simulation setting is
and the number of replication is 10,000.
Figure 1 presents the simulated results: Tc stands for τ1c and Tchat stands
ˆ ˆ
for τ2c. The rst and second columns show the simulated histograms of β τ1c β τ2c
and stopping times τ1c τ2c after normalization, and the blue curves are the
density of standard normal distribution. We could see that both sequential
estimators βˆ τ1c, βˆ τ2c and stopping times τ1c, τ2c are well approximated by
standard normal distribution. From the histograms in the third column, it’s
obvious that the variance of τ1c is larger than τ2c.

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