Solutions of Yule-Walker equations for singular AR processes

A study is presented on solutions of the Yule-Walker equations for singular AR processes that are stationary outputs of a given AR system. If the Yule-Walker equations admit more than one solution and the order of the AR system is no less than two, the solution set includes solutions which define unstable AR systems. The solution set also includes one solution, the minimal norm solution, which defines an AR system whose characteristic polynomial has either only stable zeros (implying that only one stationary output exists for this system and it is linearly regular) or has stable zeros as well as zeros of unit modulus, (implying that stationary solutions of this system are a sum of a linearly regular process and a linearly singular process). The numbers of stable and unit circle zeros of the characteristic polynomial of the defined AR system can be characterized in terms of the ranks of certain matrices, and the characteristic polynomial of the AR system defined by the minimal norm solution has the least number of unit circle zeros and the most number of stable zeros over all possible solutions.