The name is absent

Under Hi

n ^1/2B(m) → c(m)

{∑∫=1 ^Lj(1, O) f∞_oo [(f_j(s) - g_j(s,^α*)) w_j(s) exp (m_jΦ(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 K_n(a,m) = n ^1/4 ɪɪi (y_t — g(x_t,a')') W(¾m). Then, the mean value
theorem gives

K_n(a,m) — K_n(a_o,m) = n ^1/2K_n(a(m),m)n^1/4 (a — a_o) , (Al)

where sup_meM ∣∣α(m) — a_o∣∣ ≤ ∣∣a — a_o∣∣ = o_p(1). Also let

^J ρ∞

A(a_o, m)

^xj(1,O) / <⅛j(

s, a_oj)wj(s) exp (m_j∙Φ(s)) ds.

j=1 ^{7 c}

sup
meM

n ¹¹K_n(a(m),m) — A(a_o,m)

= ⅛(¹),

(A2)

by Theorem 3.2 of Park and Phillips (2001). By (Al) and (A2) we therefore have

sup ∖∣K_n(a,m) — K_n(a_o,m) + A(a_o,m)n^1/4 (a — a_o)∣∣ = o_p(1). (A3)

meM

Also note that

sup
meM

n^1/4 (a — a_o) — C(a_o)^-¹n^-^1/4 g(x_t, a_o)u_t

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)∣ = o_p(1),
meM

by Theorem 3.2 of Park and Phillips (2001). Now the result follows by (A5) and (A6)
and the assumption that inf_meM ς²(m) > O.

(ii) By Theorem 8.2 of Billingsley (1968), the following conditions are sufficient
for the requisite result: