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

In the Karlin infinite occupancy scheme, balls are thrown independently into an infinite array of boxes $1$, $2,\dots$, with probability $p_k$ of hitting the box $k$. For $j,n\in \mathbb{N}$, denote by ${\mathcal{K}}_j^{\ast }(n)$ the number of boxes containing exactly $j$ balls provided that $n$ balls have been thrown. We call $\mathit{\text{small}}\mathit{\text{counts}}$ the variables ${\mathcal{K}}_j^{\ast }(n)$, 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 ${\sum }_{k\ge 1}{\mathbb{1}}_{A_k(t)}$ as $t\to \mathrm{\infty }$, where the family of events $(A_k(t){)}_{t\ge 0}$ 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 $(A_k(t){)}_{t\ge 0}$ forms a nondecreasing family of events.