The invariance principle for a class of dependent random fields. (Russian, English) Zbl 1074.60061

Poryvaj, D. V.

Teor. Jmovirn. Mat. Stat. 70, 110-120 (2004); translation in Theory Probab. Math. Stat. 70, 123-134 (2005).

Let \(X=\{X_j,j\in\mathbb Z^d\}\) be a symmetric stationary random field. The author deals with processes of the form \(Z_n(A)=n^{-d/2}\sum_{j\in \mathbb Z^d}b_{nj}(A)X_j,A\in\mathcal U,n\in\mathbb N,\) where \(\mathcal U\) is a class of Borel subsets of the unit cube \([0,1]^d\), \(b_{nj}(A)=m((nA)\cap C_j)\), \(C_j\) is a unit cube \(C_j=(j_1-1,j_1]\times\cdots\times(j_d-1,j_d]\), \(m(\cdot)\) is the Lebesgue measure on \(\mathbb R^d\). He proposes conditions in terms of \(\beta\)-mixing coefficients under which the set of distributions \(\{Z_n(A)\colon A\in{\mathcal U}\}\) is dense in the space \(C(\overline{\mathcal U})\) of continuous functions on \(\overline{\mathcal U}\), where \(\overline{\mathcal U}\) is the closure of \({\mathcal U}\) with respect to the pseudometric \(d_L(A,B)=| A\Delta B|,A,B\in {\mathcal U}\). Conditions under which the invariance principle holds true are presented.

MSC:

60G60	Random fields
60F17	Functional limit theorems; invariance principles

Keywords:

symmetric random field