The Bernoulli sieve revisited

Abstract

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 K_n of occupied boxes, the number K_{n, 0} of empty boxes whose index is at most K_n^*, the index W_n of the first empty box and the number of balls Z_n 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 K_n and W_n under a side condition that prevents occurrence of very small boxes. The condition also ensures that K_{n, 0} converges in distribution. Limiting results for Z_n are established under an assumption of regular variation.