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Gaussian quasimartingales

  • Naresh C. Jain
  • Ditlev Monrad
Article

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 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.

Keywords

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.

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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

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