On the number of empty boxes in the Bernoulli sieve

TL;DR

This paper investigates the asymptotic behavior of the number of empty boxes in the Bernoulli sieve, revealing convergence to a functional of an inverse stable subordinator under regular variation assumptions.

Contribution

It establishes the weak convergence of the normalized number of empty boxes to a functional of an inverse stable subordinator and explores related perturbed random walk results.

Findings

Weak convergence of L_n to a functional of an inverse stable subordinator

Limiting law of L_n is mixed Poisson when convergence occurs without normalization

Derived new results for general perturbed random walks

Abstract

The Bernoulli sieve is the infinite "balls-in-boxes" occupancy scheme with random frequencies $P_k=W_1...W_{k-1}(1-W_k)$, where $(W_k)_{k\in\mn}$ are independent copies of a random variable $W$ taking values in $(0,1)$. Assuming that the number of balls equals $n$, let $L_n$ denote the number of empty boxes within the occupancy range. The paper proves that, under a regular variation assumption, $L_n$, properly normalized without centering, weakly converges to a functional of an inverse stable subordinator. Proofs rely upon the observation that $(\log P_k)$ is a perturbed random walk. In particular, some results for general perturbed random walks are derived. The other result of the paper states that whenever $L_n$ weakly converges (without normalization) the limiting law is mixed Poisson.