A poisson limit law for a generalized birthday problem

✍ Scribed by Norbert Henze

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 176 KB
• Volume: 39
• Category: Article
• ISSN: 0167-7152
• DOI: 10.1016/s0167-7152(98)00076-5

✦ Synopsis

Balls are placed sequentially at random into n cells. Write T,(~ ) for the number of balls needed until for the mth time a ball is placed into a cell already containing e-1 balls, where m/> 1 and c >_-2 are fixed integers. For fixed t >0, let X,,c denote the number of cells containing at least c balls after the placement of k, = [n l-lIe, t] balls, It is shown that, as n--+ 2, the limit distribution of X,,c is Poisson with parameter tC/c! As a consequence, the limit law of nl-C(T( ))C/c! is a Gamma distribution.