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Probability Theory and Related Fields

, Volume 92, Issue 1, pp 21–39 | Cite as

Size-biased sampling of Poisson point processes and excursions

  • Mihael Perman
  • Jim Pitman
  • Marc Yor
Article

Summary

Some general formulae are obtained for size-biased sampling from a Poisson point process in an abstract space where the size of a point is defined by an arbitrary strictly positive function. These formulae explain why in certain cases (gamma and stable) the size-biased permutation of the normalized jumps of a subordinator can be represented by a stickbreaking (residual allocation) scheme defined by independent beta random variables. An application is made to length biased sampling of excursions of a Markov process away from a recurrent point of its statespace, with emphasis on the Brownian and Bessel cases when the associated inverse local time is a stable subordinator. Results in this case are linked to generalizations of the arcsine law for the fraction of time spent positive by Brownian motion.

Keywords

Brownian Motion Markov Process Local Time General Formula Mathematical Biology 
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 1992

Authors and Affiliations

  • Mihael Perman
    • 1
  • Jim Pitman
    • 2
  • Marc Yor
    • 3
  1. 1.Department of EconomicsUniversity of LjubljanaLjubljanaYugoslavia
  2. 2.Department of StatisticsUniversity of CaliforniaBerkeleyUSA
  3. 3.Laboratoire de ProbabilitésUniversité Pierre et Marie CurieParis Cedex 05France

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