On the spectrum of correlation autoregressive sequences (Q1275942)

A sequence \(\{x_n\}\) of random variables with \(Ex_n=0\), \(E| x_n| ^2<\infty \) is called correlation autoregressive (CAR) if there exist numbers \(a_1,\dots ,a_r\) such that the covariance function \(R(m,n)=Ex_m \bar x_n\) satisfies \(R(m,n)=\sum _{k=1}^r a_k R(m+k,n+k)\) for all \(m\) and \(n\). The polynomial \(p(z)=1-\sum _{k=1}^r a_k z^k\) is called admissible. It is easy to show that any stationary sequence and any periodically correlated sequence are CAR. Examples demonstrate that \(a_k\)'s and \(r\) are not uniquely determined. However, an admissible polynomial of the lowest degree (called a minimal admissible polynomial -- MAP) is given uniquely. The authors derive a representation of \(R(n+p,n)\) based on roots of the MAP. They prove that some properties of \(\{x_n\}\) such as boundedness depend also on roots of the MAP. A spectrum of a CAR sequence is defined and a characterization of the spectrum of a bounded CAR sequence is given.