Probabilistic Number Theory, the GEM/Poisson-Dirichlet Distribution and the Arc-sine Law

Combinatorics Probability and Computing

Volumn 6, Issue 1, 1997, Pages 57-77

  Hirth, Ulrich Martin ^a,b  

^a UNIVERSITY OF LONDON   (United Kingdom)

^b CATHOLIC UNIVERSITY OF EICHSTÄTT INGOLSTADT   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031538349     PISSN: 09635483     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0963548396002805     Document Type: Article

References (26)

1. Arratia R., Barbour, A. D. and Tavaré, S. (1992) Poisson process approximations for the Ewens sampling formula. Ann. Appl. Prob. 2/3 519-535.
2. Barbour, A. D., Hoist, L. and Janson, S. (1992) Poisson approximation, Clarendon Press, Oxford.
3. Bingham, N. H. (1994) The strong arc-sine law in higher dimensions. In: Bergelson, V., March, P. and Rosenblatt, J. (eds.), Convergence in Ergodic Theory and Probability, de Gruyter, Berlin.
4. Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987) Regular variation. Encyclopedia of Mathematics and its Applications 27, Cambridge University Press (2nd ed. 1989).
5. Bingham, N. H. and Rogers, L. C. G. (1991) Summability methods and almost-sure convergence. In: Bellow, A. and Jones, R. L. (eds.), Almost Everywhere Convergence II, Academic Press, New York, 69-83.
6. Deshouillers, J.-M., Dress, F. and Tenenbaum, G. (1979) Lois de répartition des diviseurs,1. Acta Arith. 23 273-285.
7. Dickman, K. (1930) On the frequency of numbers containing prime factors of a certain relative magnitude. Ark. Mat. Astr. Fys. 22 A/10 1-14.
8. Donnelly, P. and Grimmett, G. (1993) On the asymptotic distribution of large prime factors. J. London Math. Soc. (2) 47 395-404.
9. Donnelly, P. and Tavaré, S. (1987) The population genealogy of the infinitely-many neutral alleles model. J. Math. Biol. 25 381-391.
10. Driver, R. D. (1977) Ordinary and Delay Differential Equations, Springer-Verlag, New York.
11. Hardy, G. H. and Wright, E. M. (1938) An Introduction to the Theory of Numbers, Clarendon Press, Oxford.
12. Hensley, D. (1986) The convolution powers of the Dickman function. J. London Math. Soc. (2) 33/3 395-406.
13. Hildebrand, A. (1990) The asymptotic behaviour of the solutions of a class of differential-difference equations. J. London Math. Soc. (2) 42(1) 11-31.
14. Hirth, U. M. From GEM back to Dirichlet via Hoppe's urn. Combinatorics, Probability and Computing, to appear.
15. Hirth, U. M. A Poisson approximation for Dirichlet and GEM laws by the Stein-Chen coupling method. Submitted.
16. Ignatov, Z. (1978) Point processes generated by order statistics and their applications. In: Bártfai, P. and Tomkó, J. (eds.), Point Processes and Queueing Problems, Keszthely (Hungary) (Coll. Mat. Soc. János Bolyai 24), North Holland, Amsterdam, 109-116.
17. Kemeny, J. G. (1993) Largest prime factor. J. Pure Appl. Algebra 89 81-186.
18. Kingman, J. F. C. (1993) Poisson Processes, Clarendon Press, Oxford.
19. Knuth, D. E. and Trabb Pardo, L. (1976) Analysis of a simple factorization algorithm. Theor. Comp. Sci. 3 321-348.
20. Landau, E. G. H. (1909) Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig, and Chelsea, New York.
21. Lloyd, C. J. and Williams, E. J. (1988) Recursive splitting of an interval when the proportions are identical and independent random variables. Stoch. Processes Appl. 27 111-122.
22. Perman, M., Pitman, J. and Yor, M. (1992) Size-biased sampling of Poisson point processes and excursions. Probab. Theory Relat. Fields 92 21-39.
23. Tavaré, S. (1987) The birth process with immigration, and the genealogical structure of large populations. J. Math. Biol. 25 161-168.
24. Tenenbaum, G. (1990) Introduction à la théorie analytique et probabiliste des nombres, Institut Elie Cartan No. 13, Université de Nancy 1.
25. Vershik, A. M. and Shmidt, A. A. (1977) Limit measures arising in the asymptotic theory of symmetric groups, I. Th. Prob. Appl. 22(1) 70-85.
26. Vershik, A. M. and Shmidt, A. A. (1978) Limit measures arising in the asymptotic theory of symmetric groups, II. Th. Prob. Appl. 23(1) 36-49.