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On the lengths of excursions of some Markov processes

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Séminaire de Probabilités XXXI

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 1655))

Abstract

Results are obtained regarding the distribution of the ranked lengths of component intervals in the complement of the random set of times when a recurrent Markov process returns to its starting point. Various martingales are described in terms of the Lévy measure of the Poisson point process of interval lengths on the local time scale. The martingales derived from the zero set of a one-dimensional diffusion are related to martingales studied by Azéma and Rainer. Formulae are obtained which show how the distribution of interval lengths is affected when the underlying process is subjected to a Girsanov transoformation. In particular, results for the zero set of an Ornstein-Uhlenbeck process or a Cox-Ingersoll-Ross process are derived from results for a Brownian motion or recurrent Bessel process, when the zero set is the range of a stable subordinator.

Research supported in part by N.S.F. Grant DMS-9404345

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Authors and Affiliations

  1. Department of Statistics, University of California, 367 Evans Hall #3860, 94720-3860, Berkeley, CA, USA

    Jim Pitman

  2. Laboratoire de Probabilités, Université Paris VI, 4 Place Jussieu, 75252, Paris, France

    Marc Yor

Authors
  1. Jim Pitman
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  2. Marc Yor
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Editor information

Jacques Azéma Marc Yor Michel Emery

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© 1997 Springer-Verlag Berlin Heidelberg

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Pitman, J., Yor, M. (1997). On the lengths of excursions of some Markov processes. In: Azéma, J., Yor, M., Emery, M. (eds) Séminaire de Probabilités XXXI. Lecture Notes in Mathematics, vol 1655. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0119313

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  • DOI: https://doi.org/10.1007/BFb0119313

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  • Print ISBN: 978-3-540-62634-3

  • Online ISBN: 978-3-540-68352-0

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