Electronic Journal of Probability

Limit Distributions and Random Trees Derived from the Birthday Problem with Unequal Probabilities

Michael Camarri and Jim Pitman

Full-text: Open access

Abstract

Given an arbitrary distribution on a countable set, consider the number of independent samples required until the first repeated value is seen. Exact and asymptotic formulae are derived for the distribution of this time and of the times until subsequent repeats. Asymptotic properties of the repeat times are derived by embedding in a Poisson process. In particular, necessary and sufficient conditions for convergence are given and the possible limits explicitly described. Under the same conditions the finite dimensional distributions of the repeat times converge to the arrival times of suitably modified Poisson processes, and random trees derived from the sequence of independent trials converge in distribution to an inhomogeneous continuum random tree.

Article information

Source
Electron. J. Probab., Volume 5 (2000), paper no. 2, 18 pp.

Dates
Accepted: 16 November 1999
First available in Project Euclid: 7 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1457376437

Digital Object Identifier
doi:10.1214/EJP.v5-58

Mathematical Reviews number (MathSciNet)
MR1741774

Zentralblatt MATH identifier
0953.60030

Subjects
Primary: 60G55: Point processes
Secondary: 05C05: Trees

Keywords
Repeat times point process Poisson embedding inhomogeneous continuum random tree Rayleigh distribution

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Camarri, Michael; Pitman, Jim. Limit Distributions and Random Trees Derived from the Birthday Problem with Unequal Probabilities. Electron. J. Probab. 5 (2000), paper no. 2, 18 pp. doi:10.1214/EJP.v5-58. https://projecteuclid.org/euclid.ejp/1457376437


Export citation

References

  • D. Aldous. Probability Approximations via the Poisson Clumping Heuristic, Volume 77 of Applied Mathematical Sciences. Springer-Verlag, New York, 1989.
  • D. Aldous. The continuum random tree I. Ann. Probab., 19:1-28, 1991.
  • D. Aldous. The continuum random tree III. Ann. Probab., 21:248-289, 1993.
  • D. Aldous and J. Pitman. A family of random trees with random edge lengths. Random Structures and Algorithms, 15:176-195, 1999.
  • 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.
  • A. D. Barbour, L. Holst, and S. Janson. Poisson Approximation. Clarendon Press, 1992.
  • P. Billingsley. Probability and Measure. Wiley, New York, 1995. 3rd ed.
  • A. Broder. Generating random spanning trees. In Proc. 30'th IEEE Symp. Found. Comp. Sci., pages 442-447, 1989.
  • M. Camarri. Asymptotics for $k$-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.
  • T. Carleman. Zur Theorie der Linearen Integralgleichungen. Math. Zeit, 9:196-217, 1921.
  • D. J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes. Springer-Verlag, Berlin, 1988.
  • R. Durrett. Probability: Theory and Examples. Wadsworth-Brooks/Cole, 1995. 2nd ed.
  • S. Evans and J. Pitman. Construction of Markovian coalescents. Ann. Inst. Henri Poincaré, 34:339-383, 1998.
  • 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.
  • L. Holst. On birthday, collectors', occupancy and other classical urn problems. International Statistical Review, 54:15 - 27, 1986.
  • 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).
  • J. Jaworski. On a random mapping $(T,P_j)$. J. Appl. Probab., 21:186 - 191, 1984.
  • K. Joag-Dev and F. Proschan. Birthday problem with unlike probabilities. Amer. Math. Monthly, 99:10 - 12, 1992.
  • A. Joyal. Une théorie combinatoire des séries formelles. Adv. in Math., 42:1-82, 1981.
  • R. Lyons and Y. Peres. Probability on Trees and Networks. Book in preparation, 1996.
  • 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.
  • A. Meir and J. Moon. The distance between points in random trees. J. Comb. Theory, 8:99-103, 1970.
  • 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.
  • 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.
  • J. Pitman. Coalescent random forests. J. Comb. Theory A., 85:165-193, 1999.
  • A. Rényi. 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.
  • B. Simon. Functional Integration and Quantum Physics, volume 86 of Pure and applied mathematics. Academic Press, New York, 1979.
  • 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.