Closure of Linear Processes

Peter J. Bickel and Peter Bühlmann
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Peter J. Bickel: University of California
Peter Bühlmann: University of California

Journal of Theoretical Probability, 1997, vol. 10, issue 2, 445-479

Abstract: 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.

Keywords: AR process; infinitely divisible law; MA process; distinction from nonlinear process (search for similar items in EconPapers)
Date: 1997
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DOI: 10.1023/A:1022616601841

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