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Hida-Cramér Multiplicity Theory for Multiple Markov Processes and Goursat Representations

Published online by Cambridge University Press:  22 January 2016

Loren D. Pitt
Loren D. Pitt*
Affiliation:
University of Virginia
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This work grew out of an attempt to prove the false result that an n-ple Markov process in the sense of Hida (1960) or Lévy (1956a) has multiplicity one. Instead we proved the representation theorem (Theorem III. 1.) that a centered Gaussian process x(t) is n-ple Markov iff it can be written in the form

(I.1)

where is a Gaussian martingale with

(I.2) sp {x(s): s ≤ t} ≡ sp {ai(s): s ≤ t and 1 ≤ i ≤ n}

and A(t) and {ei(t)} satisfy some non-degeneracy condition. We also show (Corollary IV. 13.) that for any Gaussian martingale A(t) with simple left innovation spectrum, continuous ei(t) may be found so that the process x(t) given in (I.1) will satisfy (I.2).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 57 , June 1975 , pp. 199 - 228
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1975

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