In a recent issue, The New Scientist ran a cover story under the title: "Mission improbable. How to predict the unpredictable"; see Matthews [448]. In it, the author describes a group of mathematicians who claim that extreme value theory (EVT) is capable of doing just that: predicting the occurrence of rare events, outside the range of available data. All members of this group, the three of us included, would immediately react with: "Yes, but, ...", or, "Be aware...". Rather than at this point trying to explain what EVT can and cannot do, we would like to quote two members of the group referred to in [448]. Richard Smith said, "There is always going to be an element of doubt, as one is extrapolating into areas one doesn't know about.

Reader Guidelines

1 Risk Theory

 1.1 The Ruin Problem

 1.2 The Cramer-Lundberg Estimate

 1.3 Ruin Theory for Heavy-Tailed Distributions

1.3.1 Some Preliminary Results

1.3.2 Cramer-Lundberg Theory for Subexponential Distributions

1.3.3 The Total Claim Amount in the Subexponential Case

 1.4 Cramer-Lundberg Theory for Large Claims: a Discussion

1.4.1 Some Related Classes of Heavy-Tailed Distributions

1.4.2 The Heavy-Tailed Cramer-Lundberg Case Revisited

2 Fluctuations of Sums

 2.1 The Laws of Large Numbers

 2.2 The Central Limit Problem

 2.3 Refinements of the CLT

 2.4 The Functional CLT: Brownian Motion Appears

 2.5 Random Sums

2.5.1 General Randomly Indexed Sequences

2.5.2 Renewal Counting Processes

2.5.3 Random Sums Driven by Renewal Counting Processes

3 Fluctuations of Maxima

 3.1 Limit Probabilities for Maxima

 3.2 Weak Convergence of Maxima Under Affine Tranformations

 3.3 Maximum Domains of Attracion and Norming Constants

3.3.1 The Mazimum Domain of Attraction of the Frechet Distribution

3.3.2 The Maximum Domain of Attraction of the Weibull Distribution W(x) = exp {- (-x)a }

3.3.3 The Maximum Domain o.f Attraction of the Gumbel Distribution A(x) = exp {- exp{-x} }

 3.4 The Generalised Extreme Value Distribution and the Generalised Pareto Distribution

 3.5 Almost Sure Behaviour of Maxima

4 Fluctuations of Upper Order Statistics

 4.1 Order Statistics

 4.2 The Limit Distribution of Upper Order Statistics

 4.3 The Limit Distribution of Randomly Indexed Upper Order Statistics.

 4.4 Some Extreme Value Theory for Stationary Sequences

5 An Approach to Extremes via Point Processes

 5.1 Basic Facts About Point Processes :

5.1.1 Definition and Examples

5.1.2 Distribution and Laplace Functional

5.1.3 Poisson Random Measures

 5.2 Weak Convergence of Point Processes

 5.3 Point Processes of Exceedances

5.3.1 The IID Case

5.3.2 The Stationary Case

 5.4 Applications of Point Process Methods to IID Sequences

5.4.1 Records and Record Times

5.4.2 Embedding Maxima in Extremal Processes

5.4.3 The Frequency of Records and the Growth of Record Times

5.4.4 Invariance Principle for Maxima

 5.5 Some Extreme Value Theory for Linear Processes

5.5.1 Noise in the Maximum Domain of Attraction of the Frechet Distribution

5.5.2 Subexponential Noise in the Maximum Domain of Attraction of the Gumbel Distribution A

6 Statistical Methods for Extremal Events

 6.1 Introduction

 6.2 Exploratory Data Analysis for Extremes

6.2.1 Probability and Quantile Plots

6.2.2 The Mean Excess Function

6.2.3 Gumbel's Method of Exceedances

6.2.4 The Return Period

6.2.5 Records as an Exploratory Tool

6.2.6 The Ratio of Maximum and Sum

 6.3 Parameter Estimation for the Generalised Extreme Value Distribution

6.3.1 Maximum Likelihood Estimation

6.3.2 Method of Probability-Weighted Moments

6.3.3 Tail and Quantile Estimation, a First Go

 6.4 Estimating Under Maximum Domain of Attraction Conditions

6.4.1 Introduction

6.4.2 Estimating the Shape Parameter

6.4.3 Estimating the Norming Constants

6.4.4 Tail and Quantile Estimation

 6.5 Fitting Excesses Over a Threshold

6.5.1 Fitting the GPD

6.5.2 An Application to Real Data

7 Time Series Analysis for Heavy-Tailed Processes

 7.1 A Short Introduction to Classical Time Series Analysis

 7.2 Heavy-Tailed Time Series

 7.3 Estimation of the Autocorrelation Function

 7.4 Estimation of the Power Transfer Function

 7.5 Parameter Estimation for ARMA Processes

 7.6 Some Remarks About Non-Linear Heavy-Tailed Models

8 Special Topics

 8.1 The Extremal Index

8.1.1 Definition and Elementary Properties

8.1.2 Interpretation and Estimation of the Extremal Index

8.1.3 Estimating the Extremal Index from Data

 8.2 A Large Claim Index

8.2.1 The Problem

8.2.2 The Index

8.2.3 Some Examples

8.2.4 On Sums and Extremes

 8.3 When and How Ruin Occurs

8.3.1 Introduction

8.3.2 The Cram@r-Lundberg Case

8.3.3 The Large Claim Case

 8.4 Perpetuities and ARCH Processes

8.4.1 Stochastic Recurrence Equations and Perpetuities

8.4.2 Basic Properties of ARCH Processes

8.4.3 Extremes of ARCH Processes

 8.5 On the Longest Success-Run

8.5.1 The Total Variation Distance to a Poisson Distribution

8.5.2 The Almost Sure Behaviour

8.5.3 The Distributional Behaviour

 8.6 Some Results on Large Deviations

 8.7 Reinsurance Treaties

8.7.1 Introduction

8.7.2 Probabilistic Analysis

 8.8 Stable Processes

8.8.1 Stable Random Vectors

8.8.2 Symmetric Stable Processes

8.8.3 Stable Integrals

8.8.4 Examples

 8.9 Self-Similarity

Appendix

 A1 Modes of Convergence

AI.1 Convergence in Distribution

A1.2 Convergence in Probability

A1.3 Almost Sure Convergence

A1.4 LP-Convergence

A1.5 Convergence to Types

A1.6 Convergence of Generalised Inverse Functions

 A2 Weak Convergence in Metric Spaces

A2.1 Preliminaries about Stochastic Processes

A2.2 The Spaces C [0, 11 and D[0, 1]

A2.3 The Skorokhod Space D (0, co)

A2.4 Weak Convergence

A2.5 The Continuous Mapping Theorem

A2.6 Weak Convergence of Point Processes

 A3 Regular Variation and Subexponentiality

A3.1 Basic Results on Regular Variation

A3.2 Properties of Subexponential Distributions

A3.3 The Tail Behaviour of Weighted Sums of Heavy-Tailed Random Variables

 A4 Some Renewal Theory 

References

Index

List of Abbreviations and Symbols