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arXiv · 1310.7055

Asymptotic distribution for the birthday problem with multiple coincidences, via an embedding of the collision process

R. Arratia·S. Garibaldi·J. Kilian
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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 $\sqrt{n\pi/2}$ 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.

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R. Arratia, S. Garibaldi, J. Kilian. 2013-10-26. Asymptotic distribution for the birthday problem with multiple coincidences, via an embedding of the collision process. https://doi.org/10.1002/rsa.20591

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