The Bernoulli Sieve Revisited

We consider an occupancy scheme in which “balls” are identified with n points sampled from the standard exponential distribution, while the role of “boxes” is played by the spacings induced by an independent random walk with positive and nonlattice steps. We discuss the asymptotic behavior of five quantities: the index K n of the last occupied box, the number Kn of occupied boxes, the number Kn,0 of empty boxes whose index is at most K ∗ n, the index Wn of the first empty box and the number of balls Zn in the last occupied box. It is shown that the limiting distribution of properly scaled and centered K n coincides with that of the number of renewals not exceeding logn. A similar result is shown for Kn and Wn under a side condition that prevents occurrence of very small boxes. The condition also ensures that Kn,0 converges in distribution. Limiting results for Zn are established under an assumption of regular variation.