#### Publication Date

5-2015

#### Abstract

In many real-life situations, a random quantity is a joint result of several independent factors, i.e., a {\em sum} of many independent random variables. The description of such sums is facilitated by the Central Limit Theorem, according to which, under reasonable conditions, the distribution of such a sum tends to normal. In several other situations, a random quantity is a {\em maximum} of several independent random variables. For such situations, there is also a limit theorem -- the Extreme Value Theorem. However, the Extreme Value Theorem is only valid under the assumption that all the components are identically distributed -- while no such assumption is needed for the Central Limit Theorem. Since in practice, the component distributions may be different, a natural question is: can we generalize the Extreme Value Theorem to a similar general case of possible different component distributions? In this paper, we use simple symmetries to prove that such a generalization is not possible. In other words, the task of modeling extremal events is provably more difficult than the task of modeling of joint effects of many factors.

## Comments

Technical Report: UTEP-CS-15-43