Irregular sets and central limit theorems (Q1857348)

Let \(\xi\) be a weakly dependent random field and \(Y\) another random field which is independent of \(\xi\). Also \(\xi\) satisfies a certain CLT and \(Y\) is assumed to satisfy only a certain law of large numbers. Consider a regression model \(X_{n}=\varphi(\xi_{n}, Y_{n}), \) \(n\in \mathbb{Z}^{d}\), where \(X_{n}\) is centered, \(\varphi\) satisfies certain regularity conditions. This type of regression models will be called I-decomposable and reachable. The author states and proves a central limit theorem \[ S_{N}(X)=S_{N}(\mathbb{Z}_{d} ; X) =\frac{1}{\sqrt{(2N+1)^d}}\sum_{n\in \mathbb{Z}_N}X_{n}\Rightarrow N(0,\sigma^{2}), \] where \(\sigma\) is an appropriate number and the convergence is the weak convergence of probability measures as \(N\rightarrow \infty.\) Note that \(A_{N}=A\cap [ -N, N ]^{d}\) for \(A\in \mathbb{Z}^{d}.\) The proof is based on the following idea: the problem can be reduced to the case of a finite-valued \(Y.\) If \(Y\) takes values in a finite set, \(x\) can be decomposed into a finite number of random fields observed over an AMC (asymptotically measurable collection). The collection \(\{ A^{i}: i=1,\ldots, r \}\) of subsets of \(\mathbb{Z}^{d}\) is called an AMC if \[ \lim_{N}F_{N}(n; A^{i}, A^{j})=F(n; A^{i}, A^{j})\;\forall n \in \mathbb{Z}^{d}, \quad i, j = 1, \dots, r, \] where \(F_{N}(n; A^{i}, A^{j})=(2N+1)^{-d}\)card\(\{ A^{i}_{N}\cap(n+A^{j}_{N})\}.\)