A Poisson limit law for a generalized birthday problem

Balls are placed sequentially at random into n cells. Write T-n,c((m)) for the number of balls needed until for the mth time a ball is placed into a cell already containing c - 1 balls, where m greater than or equal to 1 and c greater than or equal to 2 are fixed integers. For fixed t > 0, let X-n,X-c denote the number of cells containing at least c balls after the placement of k(n) = [n(1-1/c) . t] balls. It is shown that, as n --> infinity, the limit distribution of X-n,X-c is Poisson with parameter t(c)/c! As a consequence, the limit law of n(1-c)(T-n,c((m)))(c)/c! is a Gamma distribution. (C) 1998 Elsevier Science B.V. All rights reserved.