Levy Systems and Path Decompositions

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


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.


Markov Process Point Process Exit Time Predictable Process Path Decomposition 
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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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