Statistics of extremes and records in random sequences

Includes index.

Includes bibliography (p. 212 - 232) and references.

1. Introduction
2. The two principal models and some basic tools
3. First-passage probability
4. Extreme statistics
5. Time of the maximum and the minimum
6. Order statistics
7. Records
8. Extremes in other correlated systems
9. Conclusion and perspectives

"Extreme value statistics (EVS) and the statistics of records in a random sequence are examples of a truly interdisciplinary topic, spanning from statistics and mathematics on one side to physics of disordered systems on the other. They have tremendous importance and practical applications in a wide variety of fields, such as the climate science, finance, spin-glasses, random matrices, etc. One of the basic questions in EVS is to understand how the maximum or the minimum of a time series of size N fluctuates from one sample to another. The EVS is well understood when the entries of the time series under study are independent and identically distributed (IID) and this is the subject of the classical theory of EVS covered in many books and monographs, mostly in the statistics and mathematics literature. However, more recently, EVS started to play a very important role in statistical physics, in particular in disordered systems (spin-glasses and polymers in disordered medium), random matrices, random walks, fluctuating interfaces, etc. It turns out that in many physical systems the entries of the underlying time series are actually strongly correlated and the classical theory of EVS is no longer applicable. This has led to a plethora of activities, both in the statistical physics and mathematics communities over the last few decades. What is currently missing is a pedagogical book with examples illustrating the basic tools and techniques that can be useful to a student or a non-expert starting to work in this interesting and rapidly developing field. The purpose of this book is to provide an introductory monograph to this subject with a style adapted for a graduate student who only has basic knowledge of probability theory and statistical mechanics. In this book, we tried to present the basic idea and the tools, using two simple models of time series: (i) an IID sequence where there is no correlation between the entries and (ii) a random walk sequence, where the entries are strongly correlated. The EVS and related observables can be computed exactly for both models, as we illustrate in the book with several examples and exercises"--