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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.

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