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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Aldous, D., Pitman, J.: Brownian bridge asymptotics for random mappings. (Preprint, 1991)
Barlow, M., Pitman, J., Yor, M.: Une extension multidimensionnelle de la loi de l'arc sinus. Séminaire de Probabilités XXIII. (Lect. Notes Math., vol. 1372, pp. 294–314) Berlin Heidelberg New York: Springer 1989
Biane, P., Le Gall, J.F., Yor, M.: Un processus qui ressemble au pont Brownien. Séminaire de Probabilités XXI. (Lect. Notes Math., vol. 1247, pp. 270–275) Berlin Heidelberg New York: Springer 1987
Blackwell, D., MacQueen, J.B.: Ferguson distributions via Pòlya urn schemes. Ann. Stat.1, 353–355 (1973)
Brockett, P.L., Hudson, W.N.: Zeros of the densities of infinitely divisible measures. Ann. Probab.8, 400–403 (1980)
Daley, D.J., Vere-Jones, D.: An introduction to the theory of point processes. Berlin Heidelberg New York: Springer 1988
Donnelly, P., Joyce, P.: Continuity and weak convergence of ranked and size-biased permutations on the infinite simplex. Stochastic Processes Appl.31, 89–103 (1989)
Dynkin, E.B.: Some limit theorems for sums of independent random variables with infinite mathematical expectations. Trans. Math. Stat. Probab.1, 171–189 (1961)
Ewens, W.J.: Population genetics theory—the past and the future. In: Lessard, S. (ed.) Mathematical and statistical problems in evolution. University of Montreal Press. Montreal, 1988
Ferguson, T.S.: A Bayesian analysis of some nonparametric problems. Ann. Stat.1, 209–230 (1973)
Hoppe, F.M.: The sampling theory of neutral alleles and an urn model in population genetics. J. Math. Biol.25, 123–159 (1987)
Itô, K.: Poisson point processes attached to Markov processes. Proc. Sixth Berkeley Symp. Math. Statist. Prob. University of California Press, Berkeley, pp. 225–239, 1970
Kallenberg, O.: Splitting at backward times in regenerative sets. Ann. Probab.9, 781–799 (1981)
Kingman, J.F.C.: Random discrete distributions. J.R. Stat. Soc.37, 1–22 (1975)
Lukacs, E.: A characterization of the gamma distribution. Ann. Math. Stat.26, 319–324 (1955)
Lévy, P.: Fur certains processus stochastiques homogènes. Compos. Math.7, 283–339 (1939)
Molchanov, S.A., Ostrovski, E.: Symmetric stable processes as traces of degenerate diffusion processes. Theory of Probab. Appl.14, (1) 128–131 (1969)
McCloskey, J.W.: A model for the distribution of individuals by species in an environment. Unpublished Ph.D. thesis, Michigan State University, 1965
Patil, G.P., Taillie, C.: Diversity as a concept and its implications for random communities, Proc. 41st. I.S.I. New Delhi, 497–515 (1977)
Perman, M.: Random discrete distributions derived from subordinators. Ph.D. thesis, Dept Statistics, U.C. Berkeley, 1990
Perman, M.: Order statistics for normalised jumps of subordinators. (Preprint, 1991)
Pitman, J.: Size-biased sampling of independent sequences. (Preprint, 1991)
Pitman, J., Yor, M.: Arcsine laws and interval partitions derived from a stable subordinator. Proc. Lond. Math. Soc. (to appear)
Salisbury, T.S.: Construction of right processes from excursions. Z. Wahrscheinlichkeitstheor. Verw. Geb.73, 351–367 (1986)
Shepp, L.A., Lloyd, S.P.: Ordered cycle lengths in a random permutation. Trans. Am. Math Soc.121, 340–357 (1966)
Tavaré, S.: The birth process with immigration, and the genealogical structure of large populations. J. Math Biol.25, 161–168 (1987)
Vershik, A.M., Schmidt, A.A.: Limit measures arising in the asymptotic theory of symmetric groups I. Theory Probab. Appl.22, 70–85 (1977)
Vershik, A.M., Schmidt, A.A.: Limit measures arising in the asymptotic theory of symmetric groups II. Theory Probab. Appl.23, 36–49 (1978)
Watterson, G.A.: The stationary distribution of the infinitely many neutral alleles diffusion model. J. Appl. Probab.13, 639–651 (1976)
Williams, D.: Path decomposition and continuity of local time for one-dimensional diffusions. Proc. London Math. Soc.28, 738–768 (1974)
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
- Brownian Motion
- Markov Process
- Local Time
- General Formula
- Mathematical Biology