  Let $M_n^{(k)}$ denote the $k$th largest maximum of a sample
$(X_1,X_2,...,X_n)$ from parent $X$ with continuous distribution. Assume there
exist normalizing constants $a_n>0$, $b_n\in \mathbb{R}$ and a nondegenerate
distribution $G$ such that $a_n^{-1}(M_n^{(1)}-b_n)\stackrel{w}{\to}G$. Then
for fixed $k\in \mathbb{N}$, the almost sure convergence of
\[\frac{1}{D_N}\sum_{n=k}^Nd_n\mathbb{I}\{M_n^{(1)}\le a_nx_1+b_n,M_n^{(2)}\le
a_nx_2+b_n,...,M_n^{(k)}\le a_nx_k+b_n\}\] is derived if the positive weight
sequence $(d_n)$ with $D_N=\sum_{n=1}^Nd_n$ satisfies conditions provided by
H\"{o}rmann.
