A Poisson limit law for a generalized birthday problem

Balls are placed sequentially at random into n cells. Write T$_{n,c}$$^{(m)}$ for the number of balls needed until for the mth time a ball is placed into a cell already containing c − 1 balls, where m ⩾ 1 and c ⩾ 2 are fixed integers. For fixed t > 0, let X$_{n,c}$ denote the number of cells containing at least c balls after the placement of k$_n$ = [n$^{1-1c}$ $\cdot$ t] balls. It is shown that, as n → ∞, the limit distribution of X$_{n,c}$ is Poisson with parameter t$^c$/c! As a consequence, the limit law of n$^{1−c}$(T$_{n,c}$$^{(m)}$)$^c$/c! is a Gamma distribution.