General linear processes in Hilbert spaces and prediction

We discuss a general definition of linear processes in Hilbert spaces that takes into account the outstanding role played by this model in prediction theory. Actually this definition is based on the Wold decomposition of a weakly stationary process ( X n ) with values in a Hilbert space H. It leads...

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Bibliographic Details
Published in:Journal of statistical planning and inference 2007-03, Vol.137 (3), p.879-894
Main Author: Bosq, Denis
Format: Article
Language:English
Subjects:
Autoregressive processes in Hilbert spaces
Dominance
Hilbert spaces
Linear processes
Markov processes
Moving average processes in Hilbert spaces
Unbounded linear operators
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Summary:We discuss a general definition of linear processes in Hilbert spaces that takes into account the outstanding role played by this model in prediction theory. Actually this definition is based on the Wold decomposition of a weakly stationary process ( X n ) with values in a Hilbert space H. It leads to processes of the form X n = ε n + ∑ j = 1 ∞ λ j ( ε n - j ) , n ∈ Z , where ( ε n ) is a H-white noise and ( λ j ) a sequence of (possibly) unbounded linear operators. A necessary and sufficient condition for boundedness of λ j is given. As applications we introduce and study general H-autoregressive processes, H-moving average processes and H-Markov processes in the wide sense. Specific examples are given.
ISSN:0378-3758
1873-1171
DOI:10.1016/j.jspi.2006.06.014