Almost Sure Convergence of Extreme Order Statistics

Let $M_n^{(k)}$ denote the $k$th largest maximum of a sample $(X_1,X_2,\mathrm{.}\mathrm{.}\mathrm{.},X_n)$ 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 $a_n^{-1}(M_n^{(1)}-b_n)G$. Then for fixed $k\in \mathbb{N}$, the almost sure convergence of $$\frac{1}{D_N}\sum _{n=k}^N{d}_n\mathbb{I}\{M_n^{(1)}\le a_n{x}_1+b_n,M_n^{(2)}\le a_n{x}_2+b_n,\mathrm{.}\mathrm{.}\mathrm{.},M_n^{(k)}\le a_n{x}_k+b_n\}$$ is derived if the positive weight sequence $(d_n)$ with $D_N={\sum }_{n=1}^N{d}_n$ satisfies conditions provided by H"{o}rmann.