For a long period of time, the probability theory was meant to interpret the laws of averages but even within this theory occurrences now associated with extreme values had been viewed as accidents or surprises without regular laws. This view has changed only quite recently, and in this work we deal with some of the present stage problems of extreme value theory from the mathematical and practical points of view. The main aim of this work is to study the limiting behavior of the random maximum of np-dimensional distributed random variables, which are non-identical. The results of this study have found applications to many natural problems, e. g. , the project scheduling of military activities by PERT technique. Moreover, in this work we study the asymptotic behavior of the vector of the multivariate extremes, with random sample sizes. random sample sizes naturally arise in such topics as sequential analysis, branching processes, damage models or rarefaction of point processes and records as maxima.
