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

for some nonnegative constants wij such that
lim
s!1 sup
X
wij = 0: (16)
i2Zd j2Zd : (i;j)>s
Finally, we assume that the input process " = f"i; i 2 Z d g has uniformly
bounded second moments, i.e.,
Then, one can establish the following result.
sup
i2Zd k"ik2 < 1 (17)
Proposition 6 Under conditions (15)-(17), the random …eld Y = fYi; i 2 Z d g
given by (14) is well-de…ned and uniformly L2-NED on the random …eld " =
f"i; i 2 Z d g.
Conditions (15)-(17) are analogous to those used by Doukhan and Louhichi
(1999) for time-series Bernoulli shifts. Condition (15) is ful…lled if the functions
fi have bounded partial derivatives, i.e., j@fi=@xjj wij.
Thus, the class of NED spatial processes covers fairly large classes of linear
and nonlinear transformations of random …elds.
4 Limit Theorems
4.1 Central Limit Theorem
In the following, we introduce our CLT for real valued random …elds Y =
fYi;n; i 2 Dn; n 1g, where Z is assumed to be L2-NED on some vector-valued
-mixing random …eld " = f"i;n; i 2 Tn; n 1g with the NED coe¢ cients
f (s)g, where Dn Tn D and the lattice D satis…es Assumption 1. In the
sequel, we will use the following notation:
Sn = X
i2Dn
Yi;n;
2
n = var(Sn):
For ease of reference, we state below the de…nition of the -mixing coe¢ cients
employed in the paper.
De…nition 2 Let A and B be two -algebras of F, and let
(A; B) = sup(jP (AB) P (A)P (B)j; A 2 A; B 2 B);
For U Dn and V Dn, let n(U) = ("i;n; i 2 U) and n(U; V ) =
( n(U); n(V )). Then, the -mixing coe¢ cients for the random …eld " are
de…ned as:
(u; v; r) = sup
n
sup(
n(U; V ); jUj u; jV j v; (U; V ) r):
U;V
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