Simple tests for the validity of correlation function models on the circle (Q1265946)

Let \(\varphi(t)\) be a correlation function on the real line. For a given \(K>0\) define \(\psi(t)=\varphi(t)\), \(t\in [-K,K]\). Further let \(\varphi_0 (t)\) be the \(2K\)-periodic function such that \(\varphi_0(t)=\varphi(t)\) for \(t\in [-K,K]\). The problem if \(\psi(t)\) is a correlation function on the circle \([-K,K]\) is equivalent to the problem if \(\varphi_0(t)\) is a correlation function on the real line. The author presents an elegant sufficient condition, which is an analogue to the Pólya's criterion. A necessary condition based on derivatives of the function \(\varphi(t)\) is also introduced.