Central limit theorem for functionals of a linear process
The central limit theorem for a stationary sequence \(\eta_ t\), \(t\in {\mathbb{Z}}\), where \(\eta_ t=f(X_ t)\), is proved under the conditions \(\sum | r_{\eta}(t)| <\infty\) and \(\sum r_{\eta}(t)>0\). Here \(r_{\eta}(t)=E\eta_ 0\eta_ t\), \(X_ t=\sum_{s\in {\mathbb{Z}}}a(t- s)\xi_ s\) is a linear process, the variables \(\xi_ s\), \(s\in {\mathbb{Z}}\), are independent and identically distributed, \(f: {\mathbb{R}}\to {\mathbb{R}}\) belongs to some class of analytical functions which contains all polynomials. The proof is based on the method of semi-invariants and the expansion of f(x) in Appell polynomials which play the role of Hermite polynomials for the Gaussian process \(X_ t\).
scientific article; zbMATH DE number 3907457
scientific article