Decomposition of an autoregressive process into first order processes

Let Y n be an autoregressive process of order p . With p distinct characteristic roots, Y n can be decomposed into or expressed as a linear combination of p first order autoregressive processes. For the case of multiple characteristic roots, Y n with s < p distinct characteristic roots can be expressed as a linear combination of s first order autoregressive processes and the derivatives with respect to the parameter of the s first order processes. The parameters of the first order processes are the characteristic roots of Y n . Using this decomposition, for general stationary and unstable characteristic roots, the limiting distribution of appropriately normalized maximum likelihood estimators for the parameters of Y n are obtained. These results are new to the literature.