Archimedean copulas have a wide range of applications in probability theory and in mathematical statistics. A copula arose first in the work of Sklar in 1959. It is the function that characterizes principal connection of components in a random vector independently of marginal distributions. Archimedean copula process was introduced as a generalization of the scheme of independent random variables. The most of interesting applications of this object was in the order statistics theory and in the extreme values theory. The extreme type theorems and some characterizations of classes of sequences via properties of order statistics and record values can be improved from the class of independent random variables to the class of Archimedean copula processes. In this way, the proportional Archimedean copula process appeared as the generalization of the inline image-scheme for independent random variables. Some representations of the Archimedean copula process via independent random variables as well as MTPinline image