Asymptotic distribution for the birthday problem with multiple coincidences, via an embedding of the collision process
Probability
2016-04-13 v1
Abstract
We study the random variable B(c,n), which counts the number of balls that must be thrown into n equally-sized bins in order to obtain c collisions. The asymptotic expected value of B(1,n) is the well-known appearing in the solution to the birthday problem; the limit distribution and asymptotic moments of B(1,n) are also well known. We calculate the distribution and moments of B(c,n) asymptotically as n goes to infinity and c = O(n). Our main tools are an embedding of the collision process, realizing the process as a deterministic function of the standard Poisson process, and a central limit result by Renyi.
Keywords
Cite
@article{arxiv.1310.7055,
title = {Asymptotic distribution for the birthday problem with multiple coincidences, via an embedding of the collision process},
author = {R. Arratia and S. Garibaldi and J. Kilian},
journal= {arXiv preprint arXiv:1310.7055},
year = {2016}
}