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

The rest of the proof is the same, word-by-word, as the proof of Theorem 1 of
Jenish and Prucha (2009). Q.E.D.
Proof of Theorem 1: Since the proof is lengthy it is broken into steps.
1. Decomposition of Yi;n
For any …xed s > 0, decompose Xi;n as
where
Let
Yi;n = s i;n + s i;n
s
i;n = E(Yi;njFi;n(s)),
Sn;s = X
i2Dn
s
i;n = Yi;n
s
i;n; e Sn;s = X
i2Dn
s
i;n
s
i;n
2
n;s = V ar [Sn;s] ; e 2
h i
n;s = V ar eSn;s
Repeated use of the Minkowski inequality yields:
Observe that
and hence
j n n;sj en;s, j n en;sj n;s: (B.8)
E [E(Yi;njFi;n(s))jFi;n(m))] =
E(Yi;njFi;n(s)); m s;
E(Yi;njFi;n(m)); m < s:
s
i;n E( s i;njFi;n(m))
2
= kYi;n E[Yi;njFi;n(s)] E[Yi;njFi;n(m)] + E[(Yi;njFi;n(s))jFi;n(m)]k2 =
kYi;n E(Yi;njFi;n(m))k2 C (m); if m s;
kYi;n E(Yi;njFi;n(s))k2 C (s) C (m); if m < s:
since by de…nition the sequence (m) is non-increasing. Thus, for any …xed
s > 0, s i;n is uniformly L2-NED on " with the same NED coe¢ cients (m)
as the random …eld fYi;ng. Furthermore, as shown in the proof of Lemma B.3,
s
i;n is also uniformly L2+ bounded.
2. Bounds for the Variances of P Yi;n and P s i;n
First note that exists 0 < B < 1 such that for all n
BjDnj
2
n: (B.9)
Furthermore, in light of Assumption 2, and observing that = =(4 + 2 )
= =(2 + ) and b =(2+ ) (r) b 2(2+ ) (r) we have
1X
r d( +1) 1 b
r=1
=(2+ )
1X
r d( +1) 1 b =(4+2 ) (r)
r=1
26
1X
r=1
1X
r=1
r d( +1) 1 b 2(2+ ) (r) < 1;
r d( +1) 1 b 2(2+ ) (r) < 1: