The name is absent

In addition, define z_n (m) as

n ^1/4 ∑n=ι [A(a,m)C ¹(α)g(¾α) — W(¾m)] u_t
y/ s²(m)

The process z_n(m) is asymptotically equal to B(m). Moreover, by virtue of Assump-
tion C, z_n(m) is tight. This is formally stated in the subsequent lemma:

Lemma 5: Suppose that H₀ holds.

(i) Under Assumptions A and B', we have sup_meM B(m)

(ii) Under Assumptions A and C, z_n is tight.

Next, we report our main result. Let C(M) be the space of all continuous functions
on M equipped with the metric p(c₁,c₂) = sup_meM ∣c₁(m) — c₂(m)∣. The following
theorem shows the limit distribution of our test statistic under the null hypothesis.
Moreover, it establishes that the sup-statistic is unbounded in probability against any
I-regular alternative.

Theorem 3: Let Assumptions A, B' and C hold. Then, as n → ∞, we have:

(i) Under H₀, B(m) converges to z(m)², where z(m) is a mixed Gaussian element
of C(M) with covariance function

∑∫=ι ʃɪ ^Gj^(mi,^s)G_j^m -s)ds
√^s²(mι)s²(m₂)

where m₁,m₂ ∈ M and for i = ʃɪ, 2},

Gj(m_i, s) = L^1/2(G 0) [Aj(a_od,m_ij)C_j~¹(a_od)g_j(s, a_oj) — Wj(s) exp (m_ijΦ(s))] and
s^2(mi) = ∑_j=1 Lj(ɪ, ⁰⁾ ʃɪ [Aj^(ao,j^,mi,j^c)gj^(s,ao,j^{) — w}j⁽s^)exp^(mi,j$^(s))]^{2 d}s.

sup B(m) → sup z(m)².
mzM       mzM

sup B(m,yvn → sup c(m),
mzM           mzM

with sup_meM c(m) > 0 a.s.

For stationary models (e.g. Bierens (1990), de Jong (1996)), the limit distribution of
B(m) under H₀ is Gaussian. In our case it is mixed Gaussian. In addition, under
stationarity the limit distribution of the sup-statistic is data dependent (see Bierens
(1990) and de Jong (1996)). This is true under the present framework as long as the
fitted model involves multiple covariates. The limit distribution depends on the local

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