The name is absent



Under Hi

n 1/2B(m) c(m)

{∑∫=1 Lj(1, O) foo [(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 (mjΦ(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:

19



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