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Size-biased sampling of Poisson point processes and excursions. (English) Zbl 0741.60037
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 state space, 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.

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
62M09 Non-Markovian processes: estimation
60J60 Diffusion processes
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