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Closure of Linear Processes

Author

Listed:
  • Peter J. Bickel

    (University of California)

  • Peter Bühlmann

    (University of California)

Abstract

We consider the sets of moving-average and autoregressive processes and study their closures under the Mallows metric and the total variation convergence on finite dimensional distributions. These closures are unexpectedly large, containing nonergodic processes which are Poisson sums of i.i.d. copies from a stationary process. The presence of these nonergodic Poisson sum processes has immediate implications. In particular, identifiability of the hypothesis of linearity of a process is in question. A discussion of some of these issues for the set of moving-average processes has already been given without proof in Bickel and Bühlmann.(2) We establish here the precise mathematical arguments and present some additional extensions: results about the closure of autoregressive processes and natural sub-sets of moving-average and autoregressive processes which are closed.

Suggested Citation

  • Peter J. Bickel & Peter Bühlmann, 1997. "Closure of Linear Processes," Journal of Theoretical Probability, Springer, vol. 10(2), pages 445-479, April.
    • Handle: RePEc:spr:jotpro:v:10:y:1997:i:2:d:10.1023_a:1022616601841
    DOI: 10.1023/A:1022616601841
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    References listed on IDEAS

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    1. S. I. Akamanam & M. Bhaskara Rao & K. Subramanyam, 1986. "On The Ergodicity Of Bilinear Time Series Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 7(3), pages 157-163, May.
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    Cited by:

    1. Simone Giannerini & Greta Goracci, 2023. "Entropy-Based Tests for Complex Dependence in Economic and Financial Time Series with the R Package tseriesEntropy," Mathematics, MDPI, vol. 11(3), pages 1-27, February.

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