Irregular sets and central limit theorems

Abstract

In previous papers we have studied the asymptotic behaviour of$S_N(A;X)=(2N+1)^{-d/2}\Sigma_{n \in A_N}X_n$, where $X$ is a centred, 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. We present here an application of this technique. Consider a regression model $X_n=\varphi (\xi_,n,Y_n), n \in \mathbb {Z}^d$, where $X_n$ is centred, $\varphi$ satisfies certain regularity conditions, and $\xi$ and $Y$ are independent random fields; for any $m \in \mathbb {N}$ and $(y1,\ldots,y_m)$, the central limit theorem holds for ($\varphi (\xi ,y_1),\ldots, \xi ,y_m)$), but $Y$ satisfies only the strong law of large numbers as it applies to $(Y_m,Y_{m-n})_{m \in \mathbb{Z}^d}$, for any $n \in \mathbb{Z}^d$. Under these conditions, it is shown that the central limit theorem holds for $X$.