Gaps in discrete random samples

For a sequence of independent and identically distributed random variables with values in the set of non-negative integers we investigate the number of gaps and the length of the longest gap in the set of the first n values. We obtain necessary and sufficient conditions in terms of the tail sequence for the gaps to vanish asymptotically (almost surely or in probability) as n goes to infinity.

We also give a sufficient condition for  the length of the longest gap to tend to infinity in probability.

This is based on a joint work with Rudolf Gruebel from Leibniz Universitaet Hannover.