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On Multivariate Wide-sense Markov Processes*

Published online by Cambridge University Press:  22 January 2016

V. Mandrekar
V. Mandrekar*
Affiliation:
University of Minnesota
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The idea of multivariate wide-sense Markov processes has been recently used by F.J. Beutler [1], In his paper, he shows that the solution of a linear vector stochastic differential equation in a wide-sense Markov process. We obtain here a characterization of such processes and as its consequence obtain the conditions under which it satisfies Beutler’s equation. Furthermore, in stationary Gaussian case we show that these are precisely stationary Gaussian Markov processes studied by J. Doob [5].

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 33 , November 1968 , pp. 7 - 19
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1968

Footnotes

*)

This research was supported by U.S. Army Research Office, Grant No. DA-ARO-D-31-124-G562 and U.S. Air Force Office of Scientific Research, Grant No. AFOSR-381-65. Sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin, under Contract No.: DA-31-124-ARO-D-462.

References

[1] Beutler, F.J. (1963), Multivariate wide-sense Markov processes and prediction theory. Ann. Math. Statist. 34, 424–438.Google Scholar
[2] Cramér, H. (1961), On some classes of non-stationary processes. Proc. 4th Berkeley Sympos. Math. Statist, and Prob. II 57–77.Google Scholar
[3] Cramér, H. (1961), On the structure of purely non-deterministic processes. Ark. Mat. 4, 2-3, 249–266.Google Scholar
[4] Doob, J.L. (1953), Stochastic Processes New York.Google Scholar
[5] Doob, J.L. (1944), The elementary Gaussian processes. Annals of Math. Stat. vol. 15, 229–282.Google Scholar
[6] Hida, T. (1960), Canonical representations of Gaussian processes and their applications. Mem. Coll. Sci. A33 Kyoto. 109–155.Google Scholar
[7] Kallianpur, G. and Mandrekar, V. (1965), Multiplicity and representation theory of purely non-deterministic stochastic processes. Teor. Veroyatnost. i Primenen. X, USSR.Google Scholar
[8] Kalmykov, G.I. (1965), Correlation functions of a Gaussian Markov process. Dokl. Akad. Nauk SSSR.Google Scholar
[9] Rosenberg, M. (1964), Square integrability of matrix valued functions with respect to a non-negative definite Hermitian measure. Duke Math. J. 31.Google Scholar
[10] Wiener, N. and Masani, P. (1957), The prediction theory of multivariate stochastic processes I. Acta Math. 98, 111–149.Google Scholar
[11] Hille, E. and Phillips, R.S. (1957), Functional analysis and semigroups. American Math. Soc. Coll. Ch. X.Google Scholar
[12] Yosida, K. (1965), Functional Analysis. Springer-Verlag. 321–22.Google Scholar