= Exceedance count limit for extreme order statistics
{title2=$H_r(x)=G(x)\sum_{j=0}^{r-1}[-\log G(x)]^j/j!$}
For <independent and identically distributed random variables>, suppose normalized <sample maxima> have limiting <distribution function> $G$. At a threshold $a_nx+b_n$ with $G(x)>0$, the exceedance count has limiting <Poisson distribution> with parameter $-\log G(x)$. The $r$th largest observation lies below the threshold exactly when at most $r-1$ observations strictly exceed it, including in samples with ties. Summing the first $r$ limiting Poisson probabilities gives the displayed limit for every fixed positive integer $r$. The equivalence follows from $F(a_nx+b_n)^n\to G(x)$ and $-\log(1-p)\sim p$ as $p\downarrow0$.
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