A law of the iterated logarithm for small counts in Karlin's occupancy scheme

TL;DR

This paper establishes a law of the iterated logarithm for small counts in Karlin's occupancy scheme, providing new probabilistic bounds for the number of boxes with fixed small counts as the number of balls increases.

Contribution

It introduces a novel LIL for infinite sums of independent indicators with non-monotone events, extending previous results to more general settings.

Findings

Proves a law of the iterated logarithm for small counts in the scheme.

Develops a new LIL for sums of indicators with non-monotone events.

Uses Poissonization technique to facilitate the proof.

Abstract

In the Karlin infinite occupancy scheme, balls are thrown independently into an infinite array of boxes $1$, $2,\ldots$, with probability $p_k$ of hitting the box $k$. For $j,n\in\mathbb{N}$, denote by $\mathcal{K}^*_j(n)$ the number of boxes containing exactly $j$ balls provided that $n$ balls have been thrown. We call $\textit{small counts}$ the variables $\mathcal{K}^*_j(n)$, with $j$ fixed. Our main result is a law of the iterated logarithm (LIL) for the small counts as the number of balls thrown becomes large. Its proof exploits a Poissonization technique and is based on a new LIL for infinite sums of independent indicators $\sum_{k\geq 1}\Bbb{1}_{A_k(t)}$ as $t\to\infty$, where the family of events $(A_k(t))_{t\geq 0}$ is not necessarily monotone in $t$. The latter LIL is an extension of a LIL obtained recently by Buraczewski, Iksanov and Kotelnikova (2023+) in the situation that $(A_k(t))_{t\geq 0}$ forms a nondecreasing family of events.