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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

References

[1] Cramer, H. On the structure of purely non-determinist stochastic processes, Ark. Mat. 4 (1961), 249–266.Google Scholar
[2] Dunford, N. and Schwartz, J. Linear Operators, II, Interscience, New York (1963).Google Scholar
[3] Hida, T. Canonical representations of Gaussian processes and their applications, Mem. Coll. Sci. Univ. Kyoto, Ser. A, 33 (1960), 109–155.Google Scholar
[4] Hitsuda, M. Multiplicity of some classes of Gaussian processes, Nagoya Math. J. 52 (1973), 39–46.Google Scholar
[5] Lévy, P. Wiener’s random function and other Laplacian random functions, Proc. 2nd Berkeley Symp. Math. Stat. and Prob., Univ. Calif. Press, Berkeley (1951), 171–187.Google Scholar
[6] Lévy, P. A special problem of Brownian motion and a general theory of Gaussian random functions, Proc. 3rd. Berkeley Symp. Math. Stat. and Prob. II, Univ. Calif. Press, Berkeley (1956a), 133–175.Google Scholar
[7] Lévy, P. Sur une classe de courbes de l’espace Hilbert et sur une equation integrale non lineaire, Ann Ecole Norm. Sup 73 (1956b), 121–156.Google Scholar
[8] Mandrekar, V. On multivariate wide-sense Markov processes, Nagoya Math. J. 33 (1968), 7–19.Google Scholar
[9] Mandrekar, V. On the multiple Markov property of Lévy-Hida for Gaussian processes, Nagoya Math. J. 54 (1974), 69–78.Google Scholar
[10] Rosenberg, M. Square integrability of matrix valued functions with respect to a non-negative definite Hermitian measure, Duke Math. J. 31 (1964), 291–298.Google Scholar
[11] Rozanov, Yu. Spectral theory of multidimensional stationary random processes with discrete time, Uspehi Mat. Nauk (N.S.) 13 No. 2 (80) (1958), 93–142 Selected. transl, in Math. Stat. and Prob. I (1961), 127–132.Google Scholar