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

(b) infn jDnj 1 2 n > 0, where fDng is a sequence of …nite subsets of D such
that jDnj ! 1 as n ! 1:
(c) NED coe¢ cients satisfy P 1
r=1 rd 1 (r) < 1:
We can now state the following CLT for L2-NED random …elds.
Theorem 1 Suppose fDng is a sequence of …nite subsets such that jDnj ! 1
as n ! 1 and fTng is a sequence of subsets such that Dn Tn D of the
lattice D satisfying Assumption 1: Let Y = fYi;n; i 2 Dn; n 1g be a real valued
zero-mean random …eld that is L2-NED on a vector-valued -mixing random
…eld " = f"i;n; i 2 Tn; n 1g. Suppose Assumptions 2 and 3 hold, then
1
n Sn =) N(0; 1):
The above CLT contains the CLT for -mixing random …elds given in Jenish
and Prucha (2009) as a special case. As shown earlier, the class of NED random
…elds is an important extension of the class of -mixing random …elds, in that the
class includes important and widely considered …elds, e.g., spatial autoregressive
processes, that do not generally satisfy mixing conditions. It is for that reason
that an estimation theory limited to the class of -mixing random …elds is not
fully satisfactory; see, e.g., Gallant and White (1988) and Pötscher and Prucha
(1997) for an analogous discussion in the time series literature. We note that
Theorem 1 also contains as a special case the CLT for time series NED processes
of Wooldridge (1986), see Theorem 3.13 and Corollary 4.4.
Assumption 2 restricts the dependence structure of the input process ":
Assumptions 3(a),(b) are standard in the limit theory of mixing processes,
e.g., Wooldridge (1986), Davidson (1992), and de Jong (1997), and Jenish and
Prucha (2009).
Assumption 3(a) is satis…ed if the Yi;n are uniformly Lp-bounded for some
p > 2 + , i.e., sup n sup i2Dn kYi;nk p < 1. Assumption 3(b) is an asymptotic
negligibility condition that ensures that no single summand in‡uences disproportionately
the entire sum. Assumption 3(c) controls the size of the NED
coe¢ cients which measure the error in the approximation of Yi;n by ". Intuitively,
the approximation errors have to decline su¢ ciently fast with each
successive approximation. Assumption 3(c) is satis…ed if (r) = O(r d ) for
some > 0; i.e., (r) is of size d.
Theorem 1 can be easily extended to vector-valued …elds using the standard
Cramér-Wold device.
Corollary 1 Suppose fDng is a sequence of …nite subsets such that jDnj ! 1
as n ! 1 and fTng is a sequence of subsets such that Dn Tn D of the
lattice D satisfying Assumption 1: Let Y = fYi;n; i 2 Dn; n 1g with Yi;n 2
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