Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-11T13:15:19.171Z Has data issue: false hasContentIssue false
  • English
  • Français

On the multiple Markov property of Lévy-Hida for Gaussian processes

Published online by Cambridge University Press:  22 January 2016

V. Mandrekar
V. Mandrekar*
Affiliation:
Michigan State University, E. Lansing Mich. 48823
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The purpose of this note is to clarify relations between multiple Markov properties (MMP) defined by Lévy ([8], [9]) and Hida [5] for Gaussian processes and to extend some work in Lévy [8] and Hida [5]. In the stationary Gaussian case it has been shown ([5], [4]) that these notions of MMP coincide. Interesting examples of (non-stationary) processes satisfying MMP can be found in [5], [8].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 54 , July 1974 , pp. 69 - 78
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1974

References

[1] Cramer, H., Stochastic processes as curves in Hilbert space, Teoria veryatnosti i ee premenenie IX, 2, (1964), 193–204.Google Scholar
[2] Dolph, C. L. and Woodbury, M. A., On the relation between Green’s functions and covariances of certain stochastic processes and its application to unbiased linear predictor, Trans. Amer. Math. Soc. 72 (1952), 519–550.Google Scholar
[3] Doob, J. L., Stochastic Processes, John Wiley, New York, 1953.Google Scholar
[4] Dym, H. and McKean, H. P. Jr., Applications of De Branges spaces of integral functions to the prediction of stationary Gaussian processes, III. J. Math. 14 (1970), 299–343.Google Scholar
[5] Hida, T., Canonical representations of Gaussian processes and their applications, Mem. Coll. Sci. Kyoto A33 (1960), 109–155.Google Scholar
[6] Ince, E. L., Ordinary Differential Equations, Dover, New York, 1926.Google Scholar
[7] Karlin, S., Total positivity, Stanford University Press, Stanford, 1968.Google Scholar
[8] Lévy, P., A special problem of Brownian motion, and a general theory of Gaussian random functions, Proc. Third. Berk. Symp. Math. Stat. and Prob. II (1956), 133–175.Google Scholar
[9] Lévy, P., Fonctions linearement Markoviennes d’ordre n , Math. Japonicae 4 (1957), 113–121.Google Scholar
[10] Mandrekar, V., On multivariate wide-sense Markov processes, Nagoya Math. J. 33 (1968), 7–19.Google Scholar
[11] Mandrekar, V., On multiple Markov process of T. Hida and their prediction, Preprint 1971.Google Scholar