## Limit Distributions and Random Trees Derived From the Birthday Problem With Unequal Probabilities (1998)

Citations: | 26 - 14 self |

### BibTeX

@MISC{Camarri98limitdistributions,

author = {Michael Camarri and Jim Pitman},

title = {Limit Distributions and Random Trees Derived From the Birthday Problem With Unequal Probabilities},

year = {1998}

}

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### OpenURL

### Abstract

Given an arbitrary distribution on a countable set S 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 Research supported in part by N.S.F. Grants DMS 92-24857, 94-04345, 92-24868 and 97-03691 trials converge in distribution to an inhomogeneous continuum random tree. 1 Introduction Recall the classical birthday problem: given that each day of the year is equally likely as a possible birthday, and that birth...