Uncertainty Modeling pp 123-133 | Cite as
Modeling Extremal Events Is Not Easy: Why the Extreme Value Theorem Cannot Be As General As the Central Limit Theorem
Abstract
In many real-life situations, a random quantity is a joint result of several independent factors, i.e., a 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 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.
Notes
Acknowledgements
We acknowledge the partial support of the Center of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, Thailand. This work was also supported in part by the National Science Foundation grants HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and DUE-0926721.
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