Rates of convergence for normal approximation in incomplete coupon collection

A collector samples with replacement a set of n >= 2 distinct coupons until he has, for the first time, all the coupons with only m(n) is an element of {0, 1,..., n - 1} missing. If m(n) -> infinity and (n - m(n))/root n -> infinity as n -> infinity, then the asymptotic distribution of the standardized random number of necessary draws is normal. With a Fourier-analytic method, we give a bound for the rates of convergence in these central limit theorems.