Laws of the iterated and single logarithm for sums of independent indicators, with applications to the Ginibre point process and Karlin's occupancy scheme

TL;DR

This paper establishes laws of the iterated and single logarithm for sums of independent indicators, with applications to the Ginibre point process and Karlin's occupancy scheme, revealing how the growth rates of expectation and variance influence normalization.

Contribution

It provides new laws of the iterated and single logarithm for sums of independent indicators, extending classical results with applications to point processes and occupancy schemes.

Findings

LIL includes iterated logarithm when expectation and variance are comparable.

LIL simplifies to single logarithm when expectation grows faster than variance.

Applications demonstrate the laws for Ginibre point process and Karlin's occupancy scheme.

Abstract

We prove a law of the iterated logarithm (LIL) for an infinite sum of independent indicators parameterized by $t$ as $t\to\infty$. It is shown that if the expectation $b$ and the variance $a$ of the sum are comparable, then the normalization in the LIL includes the iterated logarithm of $a$. If the expectation grows faster than the variance, while the ratio $\log b/\log a$ remains bounded, then the normalization in the LIL includes the single logarithm of $a$ (so that the LIL becomes a law of the single logarithm). Applications of our result are given to the number of points of the infinite Ginibre point process in a disk and the number of occupied boxes and related quantities in Karlin's occupancy scheme.