Asymptotics for K-Fold Repeats in the Birthday Problem With Unequal Probabilities

In a previous paper Camarri and Pitman studied the asymptotics for repeat times in random sampling by a method of Poisson embedding. Here we extend these results to k-fold repeats and also indicate the relationships between the repeat processes of various orders. 1 Introduction The birthday problem in its classical form asks for the minimum sized group required so that the probability of at least one repeated birthday within the Research supported in part by N.S.F. Grants FD92-24857, FD94-04345, FD92-24868 and FD97-03691 group is greater than one half. The assumptions are a year of length 365 days, with each day equally likely as a birthday, and birthdays independent from person to person. It is well known that the answer is 23. A number of generalisations of this problem have been studied. (See Diaconis and Mosteller [6] for a conversational summary of some of these approaches.) Here we are interested in the sequential birthday problem as stated by Camarri and Pitman [4]. They c..