Under Hi
n 1/2B(m) → c(m)
{∑∫=1 Lj(1, O) f∞oo [(fj(s) - gj(s,α*)) wj(s) exp (mjΦ(s))] ds}
ς2(m, a*)
and in view of Theorem 1 and Lemma 3 c(m) > O a.s. and almost everywhere with
respect to A.
Proof of Lemma 5:
(i) Let Kn(a,m) = n 1/4 ɪɪi (yt — g(xt,a')') W(¾m). Then, the mean value
theorem gives
Kn(a,m) — Kn(ao,m) = n 1/2Kn(a(m),m)n1/4 (a — ao) , (Al)
where supmeM ∣∣α(m) — ao∣∣ ≤ ∣∣a — ao∣∣ = op(1). Also let
J ρ∞
A(ao, m)
xj(1,O) / <⅛j(
s, aoj)wj(s) exp (mj∙Φ(s)) ds.
j=1 7 c
Next
sup
meM
n 11Kn(a(m),m) — A(ao,m)
= ⅛(1),
(A2)
by Theorem 3.2 of Park and Phillips (2001). By (Al) and (A2) we therefore have
sup ∖∣Kn(a,m) — Kn(ao,m) + A(ao,m)n1/4 (a — ao)∣∣ = op(1). (A3)
meM
Also note that
sup
meM
n
n1/4 (a — ao) — C(ao)-1n-1/4 g(xt, ao)ut
t=1
= ⅛(1).
(A4)
Now (A3) and (A4) give
sup
meM
Kn(a,m) — Zn (m) y∕ς 2(m)
= ⅛(1).
(A5)
Next note that
(A6)
sup ∣ς2(m) — ς2(m)∣ = op(1),
meM
by Theorem 3.2 of Park and Phillips (2001). Now the result follows by (A5) and (A6)
and the assumption that infmeM ς2(m) > O.
(ii) By Theorem 8.2 of Billingsley (1968), the following conditions are sufficient
for the requisite result:
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