Extreme-Value Theory

Abstract

The statistics of waves is important in understanding the forces acting on the sea shore and for determining its evolution. Interaction among waves and winds is crucial for wave motion. Knowledge of the probability of occurrence of extreme events is necessary for designing secure structures in the sea environment. Extreme-value theory provides powerful tools to evaluate the probability of extreme events. In this chapter our aim is to collect several contributions to the theory of extreme events in order to make a self-contained exposition. We present a selection of the papers which seem best suited to our procedures, aims and tastes ([55], [56], [16], [19], [8], [29], [17]). Theorems are outlined without giving the proofs, which can be found in the quoted literature; we prefer to underline their importance for operations on the data. The chapter is divided in two parts. The first describes the method for deriving the distribution of the maxima in the case of independent random variables from the statistics of the exceedances of the time series over a certain threshold. This method is called POT (peak over threshold) and will be used in Chapter 9 to show the results for sea measurements. Section 6.1 also gives the fundamentals of the theory. The hypothesis of independent random variables is very restrictive and obliges the researcher to extract subsequences of i.i.d. variables from stationary processes, getting too few data in the case of long-range correlations. In the second part we deal with theorems and useful results for weakly dependent data. There is an introduction to each section with exact mathematical statements where the ideas are explained in simple and more intuitive terms.