Electronic Journal of Probability

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

Michael Camarri and Jim Pitman

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

This work is licensed under aCreative Commons Attribution 3.0 License.


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

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