Limit distributions and random trees derived from the birthday problem with unequal probabilities

Electronic Journal of Probability

Volumn 5, Issue , 2000, Pages 1-18

  Camarri, Michael ^a   Pitman, Jim ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Inhomogeneous continuum random tree; Point process; Poisson embedding; Rayleigh distribution; Repeat times

Indexed keywords

EID: 0002184446     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/EJP.v5-58     Document Type: Article

References (28)

1. D. Aldous. Probability approximations via the Poisson clumping heuristic, volume 77 of Applied Mathematical Sciences. Springer-Verlag, New York, 1989.
2. D. Aldous. The continuum random tree I. Ann. Probab., 19:1-28, 1991.
3. D. Aldous. The continuum random tree III. Ann. Probab., 21:248-289, 1993.
4. D. Aldous and J. Pitman. A family of random trees with random edge lengths. Random Structures and Algorithms, 15:176-195, 1999.
5. D. Aldous and J. Pitman. Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent. Technical Report 525, Dept. Statistics, U.C. Berkeley, 1998. To appear in Prob. Th. and Rel. Fields. Available via http://www.stat.berkeley.edu/users/pitman.
6. A. D. Barbour, L. Holst, and S. Janson. Poisson Approximation. Clarendon Press, 1992.
7. P. Billingsley. Probability and Measure. Wiley, New York, 1995. 3rd ed.
8. A. Broder. Generating random spanning trees. In Proc. 30’th IEEE Symp. Found. Comp. Sci., pages 442-447, 1989.
9. M. Camarri. Asymptotics for fc-fold repeats in the birthday problem with unequal probabilities. Technical Report 524, Dept. Statistics, U.C. Berkeley, 1998. Available via http://www.stat.berkeley.edu.
10. T. Carleman. Zur Theorie der Linearen Integralgleichungen. Math. Zeit, 9:196-217, 1921.
11. D. J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes. Springer-Verlag, Berlin, 1988.
12. R. Durrett. Probability: Theory and Examples. Wadsworth-Brooks/Cole, 1995. 2nd ed.
13. S. Evans and J. Pitman. Construction of Markovian coalescents. Ann. Inst. Henri Poincare, 34:339-383, 1998.
14. M. H. Gail, G. H. Weiss, N. Mantel, and S. J. O’Brien. A solution to the generalized birthday problem with application to allozyme screening for cell culture contamination. J. Appl. Probab., 16(2):242-251, 1979.
15. L. Holst. On birthday, collectors’, occupancy and other classical urn problems. International Statistical Review, 54:15-27, 1986.
16. L. Holst. The general birthday problem. Random Structures Algorithms, 6(2-3):201-208, 1995. Proceedings of the Sixth International Seminar on Random Graphs and Probabilistic Methods in Combinatorics and Computer Science, “Random Graphs’93” (Poznan, 1993).
17. J. Jaworski. On a random mapping (T,Pj). J. Appl. Probab., 21:186-191, 1984.
18. K. Joag-Dev and F. Proschan. Birthday problem with unlike probabilities. Amer. Math. Monthly, 99:10-12, 1992.
19. A. Joyal. Une theorie combinatoire des series formelles. Adv. in Math., 42:1-82, 1981.
20. R. Lyons and Y. Peres. Probability on trees and networks. Book in preparation, available at http://www.ma.huji.ac.il/∼lyons/prbtree.html, 1996.
21. S. Mase. Approximations to the birthday problem with unequal occurrence probabilities and their application to the surname problem in Japan. Ann. Inst. Statist. Math., 44(3):479-499, 1992.
22. A. Meir and J. Moon. The distance between points in random trees. J. Comb. Theory, 8:99-103, 1970.
23. J. Pitman. Abel-Cayley-Hurwitz multinomial expansions associated with random mappings, forests and subsets. Technical Report 498, Dept. Statistics, U.C. Berkeley, 1997. Available via http://www.stat.berkeley.edu/users/pitman.
24. J. Pitman. The multinomial distribution on rooted labeled forests. Technical Report 499, Dept. Statistics, U.C. Berkeley, 1997. Available via http://www.stat.berkeley.edu/users/pitman.
25. J. Pitman. Coalescent random forests. J. Comb. Theory A., 85:165-193, 1999.
26. A. Renyi. On the enumeration of trees. In R. Guy, H. Hanani, N. Sauer, and J. Schonheim, editors, Combinatorial Structures and their Applications, pages 355-360. Gordon and Breach, New York, 1970.
27. B. Simon. Functional integration and quantum physics, volume 86 of Pure and applied mathematics. Academic Press, New York, 1979.
28. C. Stein. Application of Newton’s identities to a generalized birthday problem and to the Poisson Binomial distribution. Technical Report 354, Dept. Statistics, Stanford University, 1990.