The Bernoulli sieve revisited

TL;DR

This paper analyzes the asymptotic behavior of various quantities in a Bernoulli sieve model where points are sampled from an exponential distribution and boxes are defined by a random walk's spacings, revealing their limiting distributions.

Contribution

It provides new asymptotic results for occupancy-related quantities in a Bernoulli sieve with random walk spacings, extending understanding of their limiting distributions.

Findings

Limiting distribution of scaled last occupied box index matches the number of renewals up to log n.

Distributional convergence of the number of empty boxes up to the last occupied box.

Asymptotic behavior of the number of balls in the last occupied box under regular variation assumptions.

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 $\log n$. 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.