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2000 Limit Distributions and Random Trees Derived from the Birthday Problem with Unequal Probabilities
Michael Camarri, Jim Pitman
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Electron. J. Probab. 5: 1-18 (2000). DOI: 10.1214/EJP.v5-58


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.


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Michael Camarri. Jim Pitman. "Limit Distributions and Random Trees Derived from the Birthday Problem with Unequal Probabilities." Electron. J. Probab. 5 1 - 18, 2000.


Accepted: 16 November 1999; Published: 2000
First available in Project Euclid: 7 March 2016

zbMATH: 0953.60030
MathSciNet: MR1741774
Digital Object Identifier: 10.1214/EJP.v5-58

Primary: 60G55
Secondary: 05C05

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


Vol.5 • 2000
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