where the last inequality is due to Assumption C. Therefore, ∑j=1 E (n' ' 2 ΣLι wi (xi,tΓ}
is bounded. This together with (A8) establishes (A7). B
Proof of Theorem 3: The result under H0 follows easily from Lemma 4 and Lemma
3.1 of Chang et. al. (2001). The result under H1 follows directly from Theorem 3.2
Park and Phillips (2001) and Lemma 3.1 of Chang et. al. (2001). ■
Proof of Theorem 4: (i) Note that [zn(m1),..., zn(mo)] → [z(m1),..., z(m^)] by
Theorem 3.2 of Park and Phillips (2001), for any finite d. In view of Lemma 4(ii)
this ensures that zn converges in distribution to z. Moreover, because supmeM (.)2 is
a continuous mapping from C(M) on R, the result follows.
(ii) The result under H1 follows directly from Theorem 3.2 of Park and Phillips
(2001). ■
Proof of Lemma 6: The result follows easily from Lemma A.B
Proof of Lemma 7: Same as the proof of Theorem 4 of Bierens (1990).■
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