On the coupon collector's remainder term

Each element in a finite population π is assigned a “bonus value”, i.e. a real number. Elements are selected from π by simple random sampling with replacement and with equal draw probabilities. Each time we receive a “new” element, i.e. an element which has not been previously selected, we receive the corresponding bonus. Let Wn denote the bonus sum after n selections. It is well known that Wn is approximately normally distributed under mild conditions. We give a remainder term estimate of the Berry-Esseen type for this normal distribution approximation.