A law of the iterated logarithm for the number of occupied boxes in the Bernoulli sieve

TL;DR

This paper establishes a law of the iterated logarithm for the number of occupied boxes in the Bernoulli sieve, providing asymptotic behavior insights for large sample sizes.

Contribution

It introduces a law of the iterated logarithm for the occupancy count in the Bernoulli sieve, a novel asymptotic result for this stochastic model.

Findings

Law of the iterated logarithm proven for the number of occupied boxes

Asymptotic behavior characterized as sample size grows large

Provides theoretical foundation for occupancy fluctuations in the Bernoulli sieve

Abstract

The Bernoulli sieve is an infinite occupancy scheme obtained by allocating the points of a uniform $[0,1]$ sample over an infinite collection of intervals made up by successive positions of a multiplicative random walk independent of the uniform sample. We prove a law of the iterated logarithm for the number of non-empty (occupied) intervals as the size of the uniform sample becomes large.