  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 $\log n$. 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.
