In previous papers, we studied the asymptotic behaviour of
$S_N(A,X)=(2N+1)^{-d/2}\sum_{n \in A_N} X_n,$ where $X$ is a centered,
stationary and weakly dependent random field, and $A_N=A \cap [-N,N]^d$, $A
\subset \mathbb{Z}^d$. This leads to the definition of asymptotically
measurable sets, which enjoy the property that $S_N(A;X)$ has a Gaussian weak
limit for any $X$ belonging to a certain class. Here we extend this type of
results to the case of weakly dependent triangular arrays and present an
application of this technique to regression models.