The Laws of Large Numbers

The Law of Large Numbers deals with three types of law of large numbers according to the following convergences: stochastic, mean, and convergence with probability 1. The book also investigates the rate of convergence and the laws of the iterated logarithm. It reviews measure theory, probability theory, stochastic processes, ergodic theory, orthogonal series, Huber spaces, Banach spaces, as well as the special concepts and general theorems of the laws of large numbers. The text discusses the laws of large numbers of different classes of stochastic processes, such as independent random variables, orthogonal random variables, stationary sequences, symmetrically dependent random variables and their generalizations, and also Markov chains. It presents other laws of large numbers for subsequences of sequences of random variables, including some general laws of large numbers which are not related to any concrete class of stochastic processes. The text cites applications of the theorems, as in numbers theory, statistics, and information theory. The text is suitable for mathematicians, economists, scientists, statisticians, or researchers involved with the probability and relative frequency of large numbers.

Introduction

Chapter 0. Mathematical Background

§ 0.1. Measure Theory

§ 0.2. Probability Theory

§ 0.3. Stochastic Processes

§ 0.4. Hubert and Banach Spaces

§ 0.5. Ergodic Theory

§ 0.6. Orthogonal Series

Chapter 1. Definitions and Generalities

§ 1.1. The Different Kinds of the Laws of Large Numbers

§ 1.2. General Theorems

Chapter 2. Independent Random Variables

§ 2.1. Inequalities

§ 2.2. The Three Series Theorem

§ 2.3. What are the Possible Limits?

§ 2.4. Convergence in Mean

§ 2.5. Weak Laws

§ 2.6. Estimation of the Rate of Convergence

§ 2.7. Strong Laws

§ 2.8. The Law of the Iterated Logarithm

§ 2.9. Identically Distributed Random Variables

§ 2.10. Weighted Averages

§ 2.11. Convergence to + ∞

Chapter 3. Orthogonal Random Variables

§ 3.1. Inequalities

§ 3.2. Convergence of Series and a Strong Law of Large Numbers

§ 3.3. Multiplicative Systems

§ 3.4. Special Orthogonal Sequences

Chapter 4. Stationary Sequences

§ 4.1. Stationary Sequences in the Strong Sense

§ 4.2. Strong and Weak Laws for Stationary Sequences in the Weak Sense

§ 4.3. The Estimation of Tthe Covariance Function

Chapter 5. Subsequences of Sequences of Random Variables

§ 5.1. A Conjecture of H. Steinhaus

§ 5.2. Subsequences of Stationary Sequences

§ 5.3. Subsequences of Special Orthogonal Sequences

Chapter 6. Symmetrically Dependent Random Variables and their Generalizations

§ 6.1. Symmetrically Dependent Random Variables

§ 6.2. Quasi-Independent Events

§ 6.3. Quasi-Multiplicative Systems

Chapter 7. Markov Chains

§ 7.1. Homogeneous Markov Chains

§ 7.2. Non-Homogeneous Markov Chains

§ 7.3. The Law of the Iterated Logarithm

Chapter 8. Weakly Dependent Random Variables

§ 8.1. A General Theorem on Centered Random Variables

§ 8.2. Mixing

Chapter 9. Independent Random Variables Taking Values in an Abstract Space

§ 9.1. Independent Random Variables Taking Values in a Hubert Space

§ 9.2. Independent Random Variables Taking Values in a Banach Space

Chapter 10. Sum of a Random Number of Independent Random Variables

Chapter 11. Applications

§ 11.1. Applications in Number Theory

§ 11.2. Applications in Statistics

§ 11.3. Applications in Information Theory

References

Author Index