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.
Highlights
Given an arbitrary distribution on a countable set S consider the number of independent samples required until the first repeated value is seen
The birthday problem can be approached by counting the number of matched pairs in a set
The arrival times for Xn are snTn1, snTn2, . . . so by Lemma 8 and standard theory of weak convergence of point processes (Daley and Vere-Jones [11, Theorem 9.1.VI]) it is enough to show that the processes Xn converge weakly to M
Summary
We present some of the main results of the paper, with pointers to following sections for details and further developments. For p the uniform distribution on a finite set this is equivalent to the formula of Meir and Moon [22] for the distribution of the distance between two distinct points in a uniform random tree. A central result of this paper, established, is the solution to this problem provided by the following theorem: Theorem 4 Let Rn1 be the index of the first repeated value in an i.i.d. sequence with discrete distribution whose point probabilities in non-increasing order are (pni, i ≥ 1). A corollary of Theorem 14, presented, describes a sense in which the sequence of random trees T (Ynj, j ≥ 0) converges in distribution in the same limit regime (9) to a continuum random tree (CRT) which can be constructed directly from the point processes in the plane. See Aldous-Pitman [4] for the study of various distributional properties of the limiting ICRT T θ, and Aldous-Pitman [5] for the application of this ICRT to the study of a coalescent process
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
