Extremal indices in a series scheme and their applications

The concept of an extremal index of a stationary random sequence is generalized to a series scheme of identically distributed random variables with random series sizes tending to infinity in probability. The new extremal indices are introduced through two definitions generalizing the basic properties of the classical extremal index. Some useful properties of the new extremal indices are proved. The paper shows how the behavior of aggregate activity maxima on random graphs (in information network models) and the behavior of maxima of random particles scores in branching processes (in biological populations models) can be described in terms of the new extremal indices. New results on models with copulas and threshold models are obtained. The paper shows that the new indices can take different values for one system and the values greater than one.