Abstract
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 Masani (1960) and Pourahmadi (1985). As a consequence it is shown that the condition W -1∈L 1qxq 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.
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Research supported by the NSF Grant MCS-8301240 and the AFOSR, Grant F49620 82 C0009. This work was done while the author was visiting Center for Stochastic Processes, University of North Carolina, Chapel Hill
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Pourahmadi, M. Autoregressive representations of multivariate stationary stochastic processes. Probab. Th. Rel. Fields 80, 315–322 (1988). https://doi.org/10.1007/BF00356109
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DOI: https://doi.org/10.1007/BF00356109