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

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.