Stochastic function theory

  • M. M. Rao
Part of the Mathematics and Its Applications book series (MAIA, volume 342)


The main aim of this chapter is to study in some detail certain technical problems arising in the treatment of continuous parameter stochastic processes. The concepts of separability and measurability are introduced and analyzed for general classes of processes. This can be done abstractly and more rapidly through the use of lifting theory and a brief discussion of the lifting theorem is included. The existence of separable and measurable modifications under various conditions is established. We illustrate these results by proving some stochastic function theoretical results, including Kolmogorov’s criterion for sample path continuity. Then we present some convergence theorems for continuous parameter martingales, under certain Vitali conditions. As an adjunct we include a general result on the existence of projective limits of projective systems of conditional probability measures, generalizing the classical case of Tulcea’s theorem. This work prepares for many refinements of martingale theory, with stopping times to be treated in the next chapter.


Canonical Representation Compact Hausdorff Space Outer Measure Fixed Discontinuity Complete Probability Space 
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Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • M. M. Rao
    • 1
  1. 1.University of CaliforniaRiversideUSA

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