Application of copulas in the proof of the almost sure central limit theorem for the $k$th largest maxima of some random variables

Our aim is to prove the almost sure central limit theorem for the $k$th largest maxima $( M_{n}^{( k) }) $, $k=1,2,\ldots , $ of $X_{1},\ldots ,X_{n}$, $n \gt k$, where $( X_{i}) $ forms a stochastic process of identically distributed r.v.’s of continuous type, having a bounded, continuous density and such that, for any fixed $n$, the family $( X_{1},\ldots ,X_{n}) $ of r.v.’s has an Archimedean copula $C^{\varPsi }$ with the inverse function of its generator, $\varPsi ^{-1}$, satisfying the condition of complete monotonicity.