Showing papers on "Subordinator published in 1992"
TL;DR: In this paper, 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.
Abstract: 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.
392 citations
TL;DR: In this paper, the fraction of time a standard one-dimensional Brownian motion B spends positive before time t has arcsine distribution was shown to be a functionals derived from the lengths and signs of excursions of B away from 0.
Abstract: Levy discovered that the fraction of time a standard one-dimensional Brownian motion B spends positive before time t has arcsine distribution, both for a fixed time when B t ¬=;0 almost surely, and for t an inverse local time, when B t =0 almost surely. This identity in distribution is extended from the fraction of time spent positive to a large collection of functionals derived from the lengths and signs of excursions of B away from 0. Similar identities in distribution are associated with any process whose zero set is the range of a stable subordinator, for instance a Bessel process of dimension d for 0
140 citations
TL;DR: In this paper, it was shown that the packing measure problem for a subordinator X(t) is equivalent to the upper local growth problem for Y(t)=min (Y皆1, Y皆2), where Y is an independent copy of X and Y is a concave upward function.
Abstract: Precise conditions are obtained for the packing measure of an arbitrary subordinator to be zero, positive and finite, or infinite. It develops that the packing measure problem for a subordinatorX(t) is equivalent to the upper local growth problem forY(t)=min (Y
1
(t), Y
2
(t)), whereY
1 andY
2 are independent copies ofX. A finite and positive packing measure is possible for subordinators “close to Cauchy”; for such a subordinator there is non-random concave upwards function that exactly describes the upper local growth ofY (although, as is well known, there is no such function for the subordinatorX itself).
23 citations