Extreme Value Distributions for Random Coupon Collector and Birthday Problems

Take n independent copies of a strictly positive random variable X and divide each copy with the sum of the copies, thus obtaining n random probabilities summing to one. These probabilities are used in independent multinomial trials with n outcomes. Let N _n(N ^* _n) be the number of trials needed until each (some) outcome has occurred at least c times. By embedding the sampling procedure in a Poisson point process the distributions of N _n and N ^* _n can be expressed using extremes of independent identically distributed random variables. Using this, asymptotic distributions as n are obtained from classical extreme value theory. The limits are determined by the behavior of the Laplace transform of X close to the origin or at infinity. Some examples are studied in detail.