Révész / Birnbaum / Lukacs The Laws of Large Numbers

1. Auflage 2014
ISBN: 978-1-4832-6902-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 176 Seiten, Web PDF

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.

Autoren/Hrsg.

Révész, Pál

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Front Cover;1
2;The Laws of Large Numbers;4
3;Copyright Page;5
4;Table of Contents;6
5;INTRODUCTION;8
6;CHAPTER 0. MATHEMATICAL BACKGROUND;12
6.1;§ 0.1. Measure theory;12
6.2;§ 0.2. Probability theory;16
6.3;§ 0.3. Stochastic processes;21
6.4;§ 0.4. Hubert and Banach spaces;24
6.5;§ 0.5. Ergodic theory;27
6.6;§ 0.6. Orthogonal series;29
7;CHAPTER 1. DEFINITIONS AND GENERALITIES;32
7.1;§ 1.1. The different kinds of the laws of large numbers;32
7.2;§ 1.2. General theorems;35
8;CHAPTER 2. INDEPENDENT RANDOM VARIABLES;40
8.1;§ 2.1. Inequalities;40
8.2;§ 2.2. The three series theorem;42
8.3;§ 2.3. What are the possible limits?;46
8.4;§ 2.4. Convergence in mean;46
8.5;§ 2.5. Weak laws;47
8.6;§ 2.6. Estimation of the rate of convergence;54
8.7;§ 2.7. Strong laws;60
8.8;§ 2.8. The law of the iterated logarithm;68
8.9;§ 2.9. Identically distributed random variables;73
8.10;§ 2.10. Weighted averages;75
8.11;§ 2.11. Convergence to +
8;81
9;CHAPTER 3. ORTHOGONAL RANDOM VARIABLES;84
9.1;§ 3.1. Inequalities;84
9.2;§ 3.2. Convergence of series and a strong law of large numbers;87
9.3;§ 3.3. Multiplicative systems;89
9.4;§ 3.4. Special orthogonal sequences;97
10;CHAPTER 4. STATIONARY SEQUENCES;98
10.1;§ 4.1. Stationary sequences in the strong sense;98
10.2;§ 4.2. Strong and weak laws for stationary sequences in the weak sense;101
10.3;§ 4.3. The estimation of the covariance function;102
11;CHAPTER 5. SUBSEQUENCES OF SEQUENCES OF RANDOM VARIABLES;104
11.1;§ 5.1. A conjecture of H. Steinhaus;104
11.2;§ 5.2. Subsequences of stationary sequences;113
11.3;§ 5.3. Subsequences of special orthogonal sequences;114
12;CHAPTER 6. SYMMETRICALLY DEPENDENT RANDOM VARIABLES AND THEIR GENERALIZATIONS;119
12.1;§ 6.1. Symmetrically dependent random variables;119
12.2;§ 6.2. Quasi-independent events;124
12.3;§ 6.3. Quasi-multiplicative systems;128
13;CHAPTER 7. MARKOV CHAINS;130
13.1;§ 7.1. Homogeneous Markov chains;131
13.2;§ 7.2. Non-homogeneous Markov chains;132
13.3;§ 7.3. The law of the iterated logarithm;137
14;CHAPTER 8. WEAKLY DEPENDENT RANDOM VARIABLES;138
14.1;§ 8.1. A general theorem on centered random variables;138
14.2;§ 8.2. Mixing;141
15;CHAPTER 9. INDEPENDENT RANDOM VARIABLES TAKING VALUES IN AN ABSTRACT SPACE;145
15.1;§ 9.1. Independent random variables taking values in a Hilbert space;146
15.2;§ 9.2. Independent random variables taking values in
a Banach space;147
16;CHAPTER 10. SUM OF A RANDOM NUMBER OF INDEPENDENT
RANDOM VARIABLES;149
17;CHAPTER 11. APPLICATIONS;152
17.1;§ 11.1. Applications in number theory;152
17.2;§ 11.2. Applications in statistics;158
17.3;§ 11.3. Applications in information theory;167
18;REFERENCES;170
19;AUTHOR INDEX;176