On Spatial Processes and Asymptotic Inference under Near$Epoch ...

and that for 2 n = V ar( P
i2Dn Vi;n) = 0 n we have
inf
n jDnj 1 2 n = inf
n jDnj 1 0 n inf
n jDnj 1 min( n) > 0:
From this we see that under the maintained assumptions Vi;n satis…es all assumptions
P
of the CLT for scalar-valued random …elds (Theorem 1) and, there-
1 fore, n Vi;n ) N(0; 1) as claimed.
Next de…ne Xi;n = Vi;n, then by analogous arguments as above
jXi;nj jYi;nj :
From the maintained uniform L2+ integrability of jYi;nj it then follows that
kXk 2+ kY k 2+ , which shows that the 2+ moments of Xi;n can be bounded
by a constant that does not depend on . Consequently if follows from the last
inequality in the proof of part (a) of Lemma B.3 that
V ar( X
i2Dn
Xi;n) = 0 n CjDnj
where C < 1 does not depend on n and . Hence
sup
n
jDnj 1 max( n) = sup jDnj
n
1 sup
j j=1
0 n C < 1.
This proves the second claim of the lemma. Q.E.D.
Proof of Theorem 2: To prove the theorem, it su¢ ces to show that jDnj 1 P
0. First we show that for each given s > 0, the conditional mean V s
i;n =
E(Yi;njFi;n(s)) satis…es the assumptions of the L1-norm LLN of Jenish and
Prucha (2009, Theorem 3). Using the Jensen and Lyapunov inequalities gives
for all s > 0, i 2 Dn; n 1:
E V s p
i;n EfE(jYi;nj p jFi;n(s))g sup E jYi;nj
n;i2Dn
p < 1:
So, V s
i;n is uniformly Lp-bounded for p > 1 and hence uniformly integrable.
For each …xed s; V s
i;n is a measurable function of f"j;n; j 2 Tn : (i; j) sg.
Observe that under Assumption 1 there exists a …nite constant C such that the
cardinality of the set fj 2 Tn : (i; j) sg is bounded by Csd ; compare Lemma
A.1 in Jenish and Prucha (2009). Hence,
V s(1; 1; r)
and thus in light of Assumption 6(b)
1X
r=1
r d 1 V s(1; 1; r)
X2s
r=1
1; r 2s
Cs d ; Cs d ; r 2s ; r > 2s
r d 1 + '(Cs d ; Cs d )
30
1X
(r + 2s) d 1 b(r) < 1:
r=1
!
i2Dn (Yi;n EYi;n) L1