Autoregressive representations of multivariate stationary stochastic processes

Abstract

Consider a q-variate weakly stationary stochastic process {Xn} with the spectral density W. The problem of autoregressive representation of {Xn} or equivalently the autoregressive representation of the linear least squares predictor of Xn, based on the infinite past is studied. It is shown that for every W in a large class of densities, the corresponding process has a mean convergent autoregressive representation. This class includes as special subclasses, the densities studied by Masani (1960) and Pourahmadi (1985). As a consequence it is shown that the condition W-1∈Lqxq1 or minimality of {Xn} is dispensable for this problem. When W is not in this class or when W has zeros of order 2 or more, it is shown that {Xn} has a mean Abel summable or mean compounded Cesaro summable autoregressive representation.