Gaussian quasimartingales

  • Naresh C. Jain
  • Ditlev Monrad


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 L2 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.


Covariance Total Variation Stochastic Process Probability Theory Variation Component 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cambanis, S.: On some continuity and differentiability properties of paths of Gaussian processes. J. Multivariate Anal. 3, 420–434 (1973)Google Scholar
  2. 2.
    Cambanis, S., Rajput, B.S.: Some zero-one laws for Gaussian processes. Ann. Prob. 1, 304–312 (1973)Google Scholar
  3. 3.
    Dellacherie, C.: Capacités et Processus Stochastiques. New York: Springer 1972Google Scholar
  4. 4.
    Doob, J.L.: Stochastic Processes, New York: Wiley 1953Google Scholar
  5. 5.
    Dudley, R.M.: The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Func. Anal. 1, 290–330 (1967)Google Scholar
  6. 6.
    Fernique, X.: Des résultats nouveaux sur les processus Gaussiens. C. R. Acad. Sci. Paris, Sér. A–B 278, A363-A365 (1974)Google Scholar
  7. 7.
    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 1974Google Scholar
  8. 8.
    Ibragimov, I.A.: Properties of sample functions for stochastic processes and embedding theorems. Theor. Prob. Appl. 18, 442–453 (1973)Google Scholar
  9. 9.
    Ito, K., Nisio, M.: On the oscillation function of Gaussian processes. Math. Scand. 22, 209–223 (1968)Google Scholar
  10. 10.
    Ito, K., Nisio, M.: On the convergence of sums of independent Banach space valued random variables. Osaka J. Math. 5, 33–48 (1968)Google Scholar
  11. 11.
    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 1981Google Scholar
  12. 12.
    Macak, I.K.: On the β-variation of a random process. Theor. Probab. Math. Statist. 14, 113–122 (1977)Google Scholar
  13. 13.
    Orey, S.: F-Processes. 5th Berkeley Sympos. Statist. Math. Probab. II, 301–313, Univ. Calif. (1965)Google Scholar
  14. 14.
    Shilov, G.E., Gurevich, B.L.: Integral, Measure, and Derivative; A Unified Approach. New Jersey: Prentice Hall 1966Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Naresh C. Jain
    • 1
  • Ditlev Monrad
    • 2
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

Personalised recommendations