## Electronic Journal of Probability

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

#### 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

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

Rights

#### 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

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