Thus, for r =1, the statistic has a mixed normal limiting distribution for m/n → 0 as m, n →
∞.10
The theoretical results derived above provide an unfeasible limit theory, since the variance
components have to be estimated. A consistent estimator of the standardized local time is given
by
L (n- 1) rJ
11
LX,n(r,a) = 2nξnsna X 1 {∣Xi∕n4<ξn}∙
Thus an estimator of
σ4(a)
Lx(r, a)(Lχ(1 ,a) - Lχ(r, a))d
Lx (1 ,a ) a
(9)
i.e. of the quantity resulting in Theorem 1 part (i)a, is given by
∆2
∆1
LX,n(r,a) (LX,n(1 ,a) - LX,n(r,a))
σ4 ( a )-----------k ---------------------ʌd a,
L X,n (1 ,a )
(10)
where
Σi=1 1 {∣¾n-a∣<ξn}n2 (x(i+1)/n Xi/n)
∑n-1
i=1
1{∣xi∕n-a∣<ξn}
In order to implement the estimator in (10), we need to choose the interval of integration, ∆ =
(∆1, ∆2)∙ Now, if we choose ∆ too small, then we may run the risk of getting an inconsistent
estimator of the term in (9). On the other hand, if we choose ∆ too large, then for some a ∈ ∆,
LX,n(r, a) and LX,n(1, a) would be very close to zero, and the estimator in (10) will result in a ratio
of two terms approaching zero.
Of course, when computing (10) we can exclude all a ∈ ∆ for which, say, Lx,n(1, a) ≤ δn, where
δ → 0 as n →∞∙ However, devicing a data-driven procedure for choosing δ is not an easy task.
In order to avoid this problem, we instead propose below an upper bound for the critical values of
the limiting distribution in Theorem 1, parts (i)a and (i)b.
In fact, note that almost surely,
2 ∞ σ4(a)
-∞
Lχ(r, a)(Lχ(1, a) - Lχ(r, a)) d
Lx (1 ,a ) a
r
σ4(a)Lx(r, a)da ≡ 2
0
σ4(Xs)ds,
where the last equality above follows from Lemma 3 in Bandi and Phillips (2003).
Now, Barndorff-Nielsen and Shephard (2002) have shown that
l (n-1)". „
^ (X(i +1)/n - Xi/n) -→ σ σ4ds,
0
(11)
10When m = n and r =1, the statistic converges to zero in probability.