The name is absent

Proof of Theorem 1: The proof is similar to that of Bierens (1990). Suppose m_o is
suchthat J_r g(s)e^m°^φ(^s)ds = 0. Noticethat

A [{s : φ)e^m°^φ^ = 0}] = A [{s : g(s) = 0}] > 0.

By Lemma A, there is δ > 0 such that J_r [g(s)e^m°^φ(^s)] e^mφs'^,ds = 0, for 0 < ∖m∖ < δ.
Therefore, J_r q(s)e^mφs'¹ ds = 0, for 0 < ∖m — m_o∖ < δ. Hence, inf_meM,m=m_o ∖m — m_o∖ >
0. This implies that M is a set of isolated points of the real line, and therefore is
countable. It is also straightforward to show that is non-dense in R.B

Proof of Lemma 4: By Lemma 3.1 of Chang et. al. (2001) the result can be proved
along the lines of Lemma 2 in Bierens (1990). ■

Proof of Theorem 2: We start with the limit result under H₀. By Lemma 3.1
of Chang et. al. (2001), the terms A_n(a.m) and C_n(a) have well defined limits,
A(a.m), C (a) say. Set D(a.m) = A'(a.m)C ¹ (a. m) and notice that D(a.m) can be
partitioned as D(a.m) = [D₁(a₁.m₁)...... D_j(a_j.m_j)]. Let a_n, a_n be mean values of
à and a_o. By Lemma 2, Lemma 3.1 of Chang et. al. (2001) and the mean value
theorem, the numerator of B(m) rescaled by n ^1/2 is

- 2

c — g(xt.a)') W(xt,m)

Lj=ι

^j   f             ∕*∞

⁽¹. ⁰⁾ /    ^[Dj ^(mj .^ao,j ^)gj ^(s.^ao,j⁾

4_1 I         7-∞

— Wj(s) exp (mjΦ(s))]² ds ^   Wj(1)

where (W₁(1)..... W_j(1)) ~ N(0.σ²I_j) and independent of Lj(1. 0)^,s. In addition,

by Lemma 3.1 of Chang et. al. (2001), the denominator of B(m) rescaled by n

^1/2ç²(m) → ς²(m.ao)

∑ Uj⁽¹.⁰⁾ ∞ ^[Dj ⁽m.ao,j ⁾gj ⁽s.ao,j⁾ — Wj ⁽s⁾exp⁽mj T⁽s^))]2 Ls l.^(A2)
j=1 I         9^-∞                                                     J

The result follows from (A1) and (A2).

18

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