 Exceedance count limit for extreme order statistics

For independent and identically distributed random variables, suppose normalized sample maxima have limiting distribution function $G$. At a threshold $a_{n}x+b_n$ 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 $F(a_{n}x+b_n{)}^n\to G(x)$ and $-\log (1-p)\sim p$ as $p\downarrow 0$.