English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In the Karlin infinite occupancy scheme, balls are thrown independently intoan infinite array of boxes $1$, $2,\ldots$, with probability $p_k$ of hittingthe box $k$. For $j,n\in\mathbb{N}$, denote by $\mathcal{K}^*_j(n)$ the numberof 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 smallcounts as the number of balls thrown becomes large. Its proof exploits aPoissonization technique and is based on a new LIL for infinite sums ofindependent indicators $\sum_{k\geq 1}\Bbb{1}_{A_k(t)}$ as $t\to\infty$, wherethe 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.

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