Poisson processes revisited

• Autorzy: Kingman J.F.C.
• Języki publikacji: EN

Abstrakty

EN

The thesis of this paper is that a good basis for defining Poisson processes on a general state space is to assume that the mean measure satisfies a simple bisection property, that every set of finite measure can be divided into two disjoint subsets of equal measure. This assumption is weaker than those usually made, and leads to simple and concrete proofs of the basic results. As an illustration, a very general version of Rényi’s characterisation theorem is proved. The paper also gives a straightforward account of the Poisson-Dirichlet distribution.

Słowa kluczowe

EN

Poisson process; bisection property; Rényi’s characterization; Poisson-Dirichlet distribution

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2006
• Tom: Vol. 26, Fasc. 1
• Strony: 77--95
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Kingman J.F.C.

• Isaac Newton Institute for Mathematical Sciences, 20 Clarkson Road, Cambridge CB3 0EH, UK

Bibliografia

• [1] R. Arratia, A. D. Barbour and S. Таvaré, Logarithmic Combinational Structures: a Probabilistic Approach, European Mathematical Society, Zurich 2003.
• [2] P. Billingsley, Оn the distribution of large prime divisors, Period. Math. Hungar. 2 (1972), pp. 283-289.
• [3] D. Blackwell, Discreteness of Ferguson selections, Ann. Statist, 1 (1973), pp. 356-358.
• [4] P. R. Halmos, Measure Theory, van Nostrand, Princeton 1950.
• [5] E. F. Harding and D. G. Kendall (Eds.), Stochastic Geometry, Wiley, New York 1974.
• [6] J. F. C. Kingman, Poisson Processes, Oxford 1993.
• [7] J. F. C. Kingman, The Poisson-Dirichlet distribution and the frequency of large prime divisors (2004, http://www.newton.cam.ac.uk/preprints/NI04019.pdf).
• [8] P. A. P. Moran, A non-Markovian quasi-Poisson process, Studia Sci. Math. Hungar. 2 (1967), pp. 425-429.
• [9] A. Rényi, Remarks on the Poisson process, Studia Sci. Math. Hungar. 2 (1967), pp. 119-123.
• [10] G. A. Watterson, The sampling theory of selectively neutral alleles, Adv. in Appl. Probab. 6 (1974), pp. 463-488.