Limit Distributions and Random Trees Derived from the Birthday Problem with Unequal Probabilities
Michael Camarri, University of California, Berkeley
Jim Pitman, University of California, Berkeley
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.
Full text: PDF | PostScript
Copyright for articles published in this journal is retained by the authors, with first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.
The authors of papers published in EJP/ECP retain the copyright. We ask for the permission to use the material in any form. We also require that the initial publication in EJP or ECP is acknowledged in any future publication of the same article.
Before a paper is published in the Electronic Journal of Probability or Electronic Communications in Probability we must receive a hard-copy of the copyright form. Please mail it to
Philippe Carmona
Laboratoire Jean Leray UMR 6629
Universite de Nantes,
2, Rue de la Houssinière BP 92208
F-44322 Nantes Cédex 03
France
You can also send it by FAX: (33|0) 2 51 12 59 12 to the attention of Philippe Carmona. You can even send a scanned jpeg or pdf of this copyright form to the managing editor ejpecpme@math.univ-nantes.fr. as an attached file.
If a paper has several authors, the corresponding author signs the copyright form on behalf of all the authors.