Large Deviation Principles for Sequences of Maxima and Minima

Abstract

In this article, we consider sequences of i.i.d. random variables and, under suitable conditions on the (common) distribution function, we prove large deviation principles for sequences of maxima, minima and pairs formed by maxima and minima. The i.i.d. random variables can be either unbounded or bounded; in the first case maxima and minima have to be suitably normalized.

Keywords:

• Asymptotic independence
• Joint distribution of minima and maxima
• Large deviations
• Regularly varying functions
• Tail of a distribution function

Mathematics Subject Classification:

• 60F10
• 60G70
• 62G30
• 62H05

Acknowledgments

We thank Paolo Baldi for suggesting the idea to consider the inclusion in the proof of Proposition 3.3. We thank Andrea Iannuzzi for some discussions which led us to consider the distance between two sets and the inclusion (9) in the proof of Proposition 3.3. We thank Thomas Mikosch for useful discussions which led to the presentation of the content of sec. 3.2. We thank Fabrizio Durante for making as aware of Schmitz (2004).