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

• Published in: Random structures & algorithms Vol. 48; no. 3; pp. 480 - 502
• Main Authors: Arratia, Richard, Garibaldi, Skip, Kilian, Joe
• Format: Journal Article
• Language: English
• Published: Hoboken Blackwell Publishing Ltd 01-05-2016; Wiley Subscription Services, Inc
• Subjects: Algorithms; Asymptotic properties; Bins; birthday problem; chi distribution; Coincidences; collisions; Counting; Mathematical analysis; Probability; Random variables; Renyi; size bias; urn problem
• ISSN: 1042-9832; 1098-2418
• DOI: 10.1002/rsa.20591

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 nπ/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 ∞ and c = O(n). We have two main tools: an embedding of the collision process — realizing the process as a deterministic function of the standard Poisson process — and a central limit result by Rényi. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 480–502, 2016