Extremal Indices in the Series Scheme and their Applications

TL;DR

This paper extends the concept of extremal index to the series scheme of i.i.d. random variables with random series sizes, introducing new indices that describe maxima behavior in complex systems like networks and biological models.

Contribution

It introduces generalized extremal indices for the series scheme, expanding their applicability to various models and revealing new properties and behaviors.

Findings

New extremal indices can exceed one.

Indices vary for the same system, indicating diverse extremal behaviors.

Applications include random graphs, branching processes, copula, and threshold models.

Abstract

We generalize the concept of extremal index of a stationary random sequence to the series scheme of identically distributed random variables with random series sizes tending to infinity in probability. We introduce new extremal indices through two definitions generalizing the basic properties of the classical extremal index. We prove some useful properties of the new extremal indices. We show how the behavior of aggregate activity maxima on random graphs (in information network models) and the behavior of maxima of random particle scores in branching processes (in biological population models) can be described in terms of the new extremal indices. We also obtain new results on models with copulas and threshold models. We show that the new indices can take different values for the same system, as well as values greater than one.