Size-biased sampling of Poisson point processes and excursions

@article{Perman1992SizebiasedSO,
  title={Size-biased sampling of Poisson point processes and excursions},
  author={Mihael Perman and Jim Pitman and Marc Yor},
  journal={Probability Theory and Related Fields},
  year={1992},
  volume={92},
  pages={21-39}
}
SummarySome 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… 
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References

SHOWING 1-10 OF 26 REFERENCES
Arcsine Laws and Interval Partitions Derived from a Stable Subordinator
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…
Continuity and weak convergence of ranked and size-biased permutations on the infinite simplex
Poisson point processes attached to Markov processes
The notion of point processes with values in a general space was formulated by K. Matthes [4]. A point process is called Poisson. if it is a-discrete and is a renewal process. We will prove in this…
Brownian Bridge Asymptotics for Random Mappings
TLDR
A new technique is introduced, which starts by specifying a coding of mappings as walks with ± 1 steps as a nonuniform random walk, and the main result is that as n→∞ the random walk rescales to reflecting Brownian bridge.
The stationary distribution of the infinitely-many neutral alleles diffusion model
An expression is found for the stationary density of the allele frequencies, in the infinitely-many alleles model. It is assumed that all alleles are neutral, that there is a constant mutation rate,…
On random discrete distributions
It is impossible to choose at random a probability distribution on a countably infinite set in a manner invariant under permutations of that set. However,approximations to such a choice can be made…
Ordered cycle lengths in a random permutation
1. Introduction. Problems involving a random permutation are often concerned with the cycle structure of the permutation. Let tY.n be the n! permutation operators on n numbered places, and let a(X) =…
ON CHARACTERIZATION OF THE GAMMA DISTRIBUTION.
Abstract : Let X sub 1, X sub 2,... be a sequence of i.i.d. random variables and S sub n = summation, j = 1 to j = n, of X sub j. If X sub 1 has a gamma distribution, (Z sub n is identically equal to…
The birth process with immigration, and the genealogical structure of large populations
  • S. Tavaré
  • Mathematics
    Journal of mathematical biology
  • 1987
This paper studies a version of the birth and immigration process in which families are followed in the order of their appearance. This age structure is related to a number of results from population…
Ferguson Distributions Via Polya Urn Schemes
Let p be any finite positive measure on (the Borel sets of) a complete separable metric space X. We shall say that a random probability measure P* on X has a Ferguson distribution with parameter p if…
...
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