Non-classical problems of the extreme value theory

We pose and solve the following non-classical problems of the extreme value theory.

• A sufficient condition is given for the asymptotic equivalence of the maxima in a general scheme of sums of maxima of i.i.d. random variables with heavy tails; applications of these results is described in studying of maxima of the total activity in information networks models.

• New limit theorems are developed for extremes of particles scores for branching processes in non-classical statements.
• Two notion of extremal indices for arrays of random number of dependent random variables have been introduced; theirs properties are studied, as well as theirs relations with the classic notion of the extremal index .
• The notion of maximal branching processes has been introduced, by analogy of classical branching processes; main properties of such the processes are established, in particular, ergodic and limit theorems are proven; some applications are also considered.