Stationary Gaussian Markov processes as limits of stationary autoregressive time series

Abstract

We consider the class, C p , of all zero mean stationary Gaussian processes, { Y t : t ∈ ( − ∞ , ∞ ) } with p derivatives, for which the vector valued process { ( Y t ( 0 ) , … , Y t ( p ) ) : t ≥ 0 } is a p + 1 -vector Markov process, where Y t ( 0 ) = Y ( t ) . We provide a rigorous description and treatment of these stationary Gaussian processes as limits of stationary AR( p ) time series.