Size-biased sampling of Poisson point processes and excursions

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.

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Research supported in part by NSF grant DMS88-01808 and DMS91-07351

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Perman, M., Pitman, J. & Yor, M. Size-biased sampling of Poisson point processes and excursions. Probab. Th. Rel. Fields 92, 21–39 (1992). https://doi.org/10.1007/BF01205234

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Keywords

  • Brownian Motion
  • Markov Process
  • Local Time
  • General Formula
  • Mathematical Biology