Autoregressive representations of multivariate stationary stochastic processes
Consider a q-variate weakly stationary stochastic process \(\{X_ n\}\) with the spectral density W. The problem of autoregressive representation of \(\{X_ n\}\) or equivalently the autoregressive representation of the linear least squares predictor of \(X_ n\) 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 \textit{P. Masani} [Acta Math. 104, 141-162 (1960; Zbl 0096.115)] and the author [J. Multivariate Anal. 16, 265-536 (1985; Zbl 0583.62086)]. As a consequence it is shown that the condition \(W^{-1}\in L\) \(1_{q\times q}\) or minimality of \(\{X_ n\}\) 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 \(\{X_ n\}\) has a mean Abel summable or mean compounded Cesáro summable autoregressive representation.
scientific article; zbMATH DE number 4042960
scientific article