Limit Distributions for Sums of Independent Random Vectors:  Heavy Tails in Theory and Practice

 TABLE OF CONTENTS

1. Random Vectors

1.1 Probability Distributions
1.2 Convergence in Distribution
1.3 Characteristic Functions

2. Linear Operators

2.1 Operator Norms
2.2 Exponential Operators and Powers
2.3 Convergence of Types

3. Infinitely Divisible Distributions and Triangular Arrays

3.1 Infinitely Divisible Distributions
3.2 Convergence of Triangular Arrays
3.3 Domains of Attraction
3.4 Appendix: Continuous Mappings on the Circle
3.5 Notes and Comments

4. Regular Variation for Linear Operators

4.1 Definitions and Basic Properties
4.2 Uniform Convergence and Path Behavior
4.3 The Spectral Decomposition
4.4 Notes and Comments

5. Regular Variation for Real-Valued Functions

5.1 Regularly Varying Functions
5.2 Exponents and Symmetries
5.3 Uniform Regular Variation
5.4 Notes and Comments

6. Regular Variation for Borel Measures

6.1 Regularly Varying Measures
6.2 R-O Varying Measures
6.3 Truncated Moments and Tail Moments
6.4 Sharp Spectral Bounds
6.5 Notes and Comments

7. The Limit Distributions

7.1 Operator Semistable Laws
7.2 Operator Stable Laws
7.3 Stable Laws
7.4 Semistable Laws
7.5 Structure Theorems
7.6 Notes and Comments

8. Central Limit Theorems

8.1 Normal Limits
8.2 Nonnormal Limits
8.3 General Limits
        8.3.1 Stochastic Compactness
        8.3.2 Operator Stable Limits
8.4 Notes and Comments

9. Related Limit Theorems

9.1 Large Deviations
9.1 Law of the Iterated Logarithm
9.3 Convergence of Semitypes
9.4 Notes and Comments

10. Applications to Statistics

10.1 Sample Moments
10.2 Sample Covariance Matrix
10.3 Self-Normalized Sums
10.4 Tail and Moment Estimators
10.5 Symmetric k-Tensors
10.6 Time Series Analysis
10.7 Notes and Comments

11. Self-Similar Stochastic Processes

12.1 Operator-Self-Similar Processes
12.2 Scaling Limits
12.3 Notes and Comments

The central limit theory in probability and statistics treats a fundamental scientific question. If repeated experiments under controlled conditions produce different answers, how can we draw useful conclusions from our data? The answer lies in the fact that there is a regular, nonrandom pattern to the variations in repeated experiments. The most familiar example is the bell--shaped curve. The central limit theorem presented in most basic probability and statistics courses justifies use of the bell--shaped curve. It states that sums of large numbers of independent random variables must approximately fit this distribution under a broad range of realistic conditions, an approximation which becomes more precise as the quantity of data increases. This limit theorem applies as long as the tails of the distribution are not too heavy, so that the variance is finite. This means, roughly speaking, that the probability of experimental outcomes very far from average is very small. Other central limit theorems may apply when the distributional tails are so heavy that the variance is infinite. The resulting limit distributions are called stable laws. Real world problems often involve several related measurements, so that the results of repeated experiments may usefully be represented as random vectors. Then the requisite central limit theory needs to be multidimensional. In the multivariable central limit theorem the corresponding analogue of the bell--shaped curve is called a multivariate normal or Gaussian distribution. There is also a richer limit theory which pertains when the data has heavy tails. Here the limit distributions include multivariable stable and operator stable laws, along with the semistable and operator semistable laws.
 

Stable laws were once considered just a mathematical curiosity. Around 1960 researchers began to discover evidence of heavy tail fluctuations in financial data [Fama, Mandelbrot]. This line of research led to the discovery of fractals [Mandelbrot]. By now, stable models are firmly established in the area of finance. There is extensive empirical evidence of heavy tail price fluctuations in stock markets, futures markets, and currency exchange rates [Jansen and de Vries, Loretan and Phillips,McCulloch]. Heavy tail probability distributions are also used in electrical engineering [Nikias and Shao] and hydrology [Anderson and Meerschaert, Benson et al., Hosking and Wallis]. Several additional applications to economics and computer science appear in [Adler, et al.]. Applications in other areas of science are emerging rapidly, and the subject continues to gain momentum. Multivariable heavy tail models are used in finance for portfolio analysis involving several different stock issues or mutual funds [Mittnik and Rachev, Nolan et al.] and in hydrology to describe certain multivariable diffusion models [Meerschaert et al.].

This book is intended as an accessible reference which treats the general central limit theory in detail. In our own collaborations with graduate students and faculty in economics and hydrogeology, as well as researchers in probability and statistics, it has become clear that such a book could be useful. Following the example of Gnedenko and Kolmogorov, this book starts at the beginning and carefully develops the central limit theory in detail. Starting with the basic constructions of modern probability theory, we develop the fundamental tools of infinitely divisible distributions and regular variation. Then we lay out the general central limit theory for independent random vectors. Finally we provide a number of extensions and applications to probability and statistics. Our regular variation approach mirrors that of Feller, which has become a standard reference for theoreticians as well as practicing engineers and scientists. We hope that our text will also serve as a handy reference for multivariable regular variation theory.

Probability and Statistics is a diverse field of study. The central limit theory, aside from being fundamental, provides an accessible starting point. Because of the fundamental nature of the subject, we can take the motivated reader all the way from the basic foundations of probability theory to cutting-edge research, all in one volume. Anyone with a working knowledge of analysis and linear algebra should find the book accessible. Although we do not intend our book as a textbook, it could easily form the basis for a PhD level course or seminar in probability theory. For researchers, we hope that this book will provide an efficient and logical path through a large collection of results with many possible applications to real world phenomena. Heavy tail models are rapidly gaining acceptance and importance in many fields of science and engineering, yet some of the results which are used are not easily accessible in their original form as published research articles. This has led to numerous mistakes and misunderstandings. We hope that our exposition will help to clarify and unify this important and fundamental body of research.

Mark M. Meerschaert