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Levy Systems and Path Decompositions

  • J. W. Pitman
Chapter
Part of the Progress in Probability and Statistics book series (PRPR, volume 1)

Abstract

Itô [21] introduced the idea of a point process attached to a Markov process X, and subsequent work of Weil [42], Getoor [11], [12] and Maisonneuve [29] has shown that the existence of a suitably Markovian Lévy system for such a point process can be instrumental in establishing path decompositions of the Markov process. A path decomposition, or splitting time theorem, is a result to the effect that some fragment of the trajectory of X is conditionally independent of some other fragment given suitable conditioning variables, usually with one or more of the fragments being conditionally Markovian. Millar [32] gives a survey of such results, and more recent work may be found in the papers of Getoor, Pittenger, and Sharpe: [12], [14], [15], [16], [17], [18], [36], [37], [40]. Lévy systems suitable for deriving path decompositions were constructed in varying degrees of generality by Watanabe [41] and Benveniste and Jacod [2] for the point process of jumps, and by Itô [21], Dynkin [10] and Maisonneuve [28] for point processes of excursions.

Keywords

Markov Process Point Process Exit Time Predictable Process Path Decomposition 
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

© Birkhäuser Boston 1981

Authors and Affiliations

  • J. W. Pitman
    • 1
  1. 1.Department of StatisticsUniversity of California at BerkeleyBerkeleyUSA

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