Asymptotic results for silent elimination

Following the model of Bondesson, Nilsson, and Wikstrand, we consider randomly filled urns, where the probability of falling into urn i is the geometric probability (1− q)qi−1. Assuming n independent random entries, and a fixed parameter k, the interest is in the following parameters: Let T be the smallest index, such that urn T is non-empty, but the following k are empty, then: XT = number of balls in urn T , ST = number of balls in urns with index larger than T , and finally T itself. We analyse the recursions (that appeared earlier) precisely, and derive results about the joint distribution of a related urn model.