Summary
Necessary and sufficient conditions in terms of the mean function and covariance are obtained for a separable Gaussian process to have paths of bounded variation, absolutely continuous or continuous singular. If almost all paths are of bounded variation, the L 2 expansion of the Gaussian process is shown to converge in the total variation norm. One then obtains a decomposition of the paths of a Gaussian quasimartingale into a martingale and a predictable process of bounded variation paths such that these components are jointly Gaussian; the martingale component is decomposed into two processes, one consisting of (fixed) jumps and the other a continuous path martingale, and the bounded variation component is decomposed into three processes, one consisting of (fixed) jumps, another with absolutely continuous paths and the third with continuous singular paths. All components are jointly Gaussian. Uniqueness of the decompositions is also established.
Article PDF
Similar content being viewed by others
References
Cambanis, S.: On some continuity and differentiability properties of paths of Gaussian processes. J. Multivariate Anal. 3, 420–434 (1973)
Cambanis, S., Rajput, B.S.: Some zero-one laws for Gaussian processes. Ann. Prob. 1, 304–312 (1973)
Dellacherie, C.: Capacités et Processus Stochastiques. New York: Springer 1972
Doob, J.L.: Stochastic Processes, New York: Wiley 1953
Dudley, R.M.: The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Func. Anal. 1, 290–330 (1967)
Fernique, X.: Des résultats nouveaux sur les processus Gaussiens. C. R. Acad. Sci. Paris, Sér. A–B 278, A363-A365 (1974)
Fernique, X.: Regularite des trajectoires des fonctions aléatoires Gaussiennes, Ecole d'été calcul de probabilités. St. Flour, IV. Lecture Notes in Math. 480, 1–96. Berlin-Heidelberg-New York: Springer 1974
Ibragimov, I.A.: Properties of sample functions for stochastic processes and embedding theorems. Theor. Prob. Appl. 18, 442–453 (1973)
Ito, K., Nisio, M.: On the oscillation function of Gaussian processes. Math. Scand. 22, 209–223 (1968)
Ito, K., Nisio, M.: On the convergence of sums of independent Banach space valued random variables. Osaka J. Math. 5, 33–48 (1968)
Jain, N.C., Monrad, D.: Gaussian measures in certain functions spaces. Proc. Third International Conference on Probability in Banach Spaces. (Tufts University, August 1980) Lecture Notes in Math. 860, 246–256. Berlin-Heidelberg-New York: Springer 1981
Macak, I.K.: On the β-variation of a random process. Theor. Probab. Math. Statist. 14, 113–122 (1977)
Orey, S.: F-Processes. 5th Berkeley Sympos. Statist. Math. Probab. II, 301–313, Univ. Calif. (1965)
Shilov, G.E., Gurevich, B.L.: Integral, Measure, and Derivative; A Unified Approach. New Jersey: Prentice Hall 1966
Author information
Authors and Affiliations
Additional information
This work was partially supported by the National Science Foundation.
Rights and permissions
About this article
Cite this article
Jain, N.C., Monrad, D. Gaussian quasimartingales. Z. Wahrscheinlichkeitstheorie verw Gebiete 59, 139–159 (1982). https://doi.org/10.1007/BF00531739
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00531739