Limit distributions and random trees derived from the birthday problem with unequal probabilities. (English) Zbl 0953.60030

Camarri, Michael; Pitman, Jim

Electron. J. Probab. 5, Paper No. 1, 19 p. (2000).

Summary: 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.

MSC:

60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
05C05	Trees
05C80	Random graphs (graph-theoretic aspects)

Keywords:

repeat times; point process; Poisson embedding; inhomogeneous continuum random tree; Rayleigh distribution