On Beveridge-Nelson decomposition and limit theorems for linear random fields
Let \(X_t= \sum c_k\varepsilon_{t-k}\) be a linear process. The \(\varepsilon_i\) are i.i.d. random variables and \(c_k\) and \(\varepsilon_{t-k}\) correctly define the process as stationary. The sum of dependent random variables \(\sum X_t\) can be investigated using so-called Beveridge-Nelson decomposition (BND) of linear processes. Limit theorems for \(\sum X_t\) are reduced to limit theorem for \(\sum\varepsilon_t\). So it is possible to prove the law of large numbers (LLN), strong LLN (SLLN), central limit theorem (CLT) and invariance principle (IP). The BND for random fields and some properties of corresponding coefficients are formulated. The BND is directly and also indirectly applied to prove SLLN and CLT. A limit theorem for sum of \(X_{t,s}\), where \(X_{t,s}\) is a linear field and \(t\), \(s\) are elements of some increasing sequence of subsets of \(\mathbb{Z}\), is formulated.
scientific article; zbMATH DE number 5669341