Laws of the iterated and single logarithm for sums of independent indicators, with applications to the Ginibre point process and Karlin's occupancy scheme
Dariusz Buraczewski, Alexander Iksanov, Valeriya Kotelnikova
TL;DR
This work derives a law of the iterated logarithm for infinite sums of independent indicators, revealing two distinct normalization regimes based on the relative growth of the mean and variance. It builds a robust analytical framework—central limit theorems, exponential moment bounds, and partition-based controls—to handle dependence across the indicator families, and then applies these results to two rich settings: the infinite Ginibre point process and Karlin's occupancy scheme. In the Ginibre case, the LIL implies a precise a.s. fluctuation scale for the disk-count process, while in Karlin’s scheme it yields explicit LIL constants under various regular variation and de Haan-type conditions, both for Poissonized and de-Poissonized versions. The methods unify probabilistic limit theorems for infinite indicator sums with concrete applications to point processes and occupancy problems, offering a versatile template for related stochastic systems.
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.