Ash / Birnbaum / Lukacs Real Analysis and Probability

Probability and Mathematical Statistics: A Series of Monographs and Textbooks
1. Auflage 2014
ISBN: 978-1-4831-9142-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 494 Seiten, Web PDF

Real Analysis and Probability provides the background in real analysis needed for the study of probability. Topics covered range from measure and integration theory to functional analysis and basic concepts of probability. The interplay between measure theory and topology is also discussed, along with conditional probability and expectation, the central limit theorem, and strong laws of large numbers with respect to martingale theory. Comprised of eight chapters, this volume begins with an overview of the basic concepts of the theory of measure and integration, followed by a presentation of various applications of the basic integration theory. The reader is then introduced to functional analysis, with emphasis on structures that can be defined on vector spaces. Subsequent chapters focus on the connection between measure theory and topology; basic concepts of probability; and conditional probability and expectation. Strong laws of large numbers are also examined, first from the classical viewpoint, and then via martingale theory. The final chapter is devoted to the one-dimensional central limit problem, paying particular attention to the fundamental role of Prokhorov's weak compactness theorem. This book is intended primarily for students taking a graduate course in probability.

Autoren/Hrsg.

Ash, Robert B.

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Front Cover
;1
2;Real Analysis and Probability;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;10
6;Summary of Notation;12
6.1;1 Sets;12
6.2;2 Real Numbers;13
6.3;3 Functions;14
6.4;4 Topology;14
6.5;5 Vector Spaces;15
6.6;6 Zorn's Lemma;16
7;Chapter 1. Fundamentals of Measure and Integration Theory;18
7.1;1.1 Introduction;18
7.2;1.2 Fields, s-Fields, and Measures;20
7.3;1.3 Extension of Measures;30
7.4;1.4 Lebesgue-Stieltjes Measures and Distribution Functions;39
7.5;1.5 Measurable Functions and Integration;51
7.6;1.6 Basic Integration Theorems;60
7.7;1.7 Comparison of Lebesgue and Riemann Integrals;70
8;Chapter 2. Further Results in Measure and Integration Theory;75
8.1;2.1 Introduction;75
8.2;2.2 Radon-Nikodym Theorem and Related Results;80
8.3;2.3 Applications to Real Analysis;87
8.4;2.4 Lp Spaces;97
8.5;2.5 Convergence of Sequences of Measurable Functions;109
8.6;2.6 Product Measures and Fubini's Theorem;113
8.7;2.7 Measures on Infinite Product Spaces;125
8.8;2.8 References;129
9;Chapter 3. Introduction to Functional Analysis;130
9.1;3.1 Introduction;130
9.2;3.2 Basic Properties of Hilbert Spaces;133
9.3;3.3 Linear Operators on Normed Linear Spaces;144
9.4;3.4 Basic Theorems of Functional Analysis;155
9.5;3.5 Some Properties of Topological Vector Spaces;167
9.6;3.6 References;184
10;Chapter 4. The Interplay between Measure Theory and Topology;185
10.1;4.1 Introduction;185
10.2;4.2 The Daniell Integral;187
10.3;4.3 Measures on Topological Spaces;195
10.4;4.4 Measures on Uncountably Infinite Product Spaces;206
10.5;4.5 Weak Convergence of Measures;213
10.6;4.6 References;217
11;Chapter 5. Basic Concepts of Probability;218
11.1;5.1 Introduction;218
11.2;5.2 Discrete Probability Spaces;219
11.3;5.3 Independence;220
11.4;5.4 Bernoulli Trials;222
11.5;5.5 Conditional Probability;223
11.6;5.6 Random Variables;225
11.7;5.7 Random Vectors;228
11.8;5.8 Independent Random Variables;230
11.9;5.9 Some Examples from Basic Probability;233
11.10;5.10 Expectation;240
11.11;5.11 Infinite Sequences of Random Variables;248
11.12;5.12 References;252
12;Chapter 6. Conditional Probability and Expectation;253
12.1;6.1 Introduction;253
12.2;6.2 Applications;254
12.3;6.3 The General Concept of Conditional Probability and Expectation;257
12.4;6.4 Conditional Expectation Given a s-field;266
12.5;6.5 Properties of Conditional Expectation;271
12.6;6.6 Regular Conditional Probabilities;279
12.7;6.7 References;285
13;Chapter 7. Strong Laws of Large Numbers and Martingale Theory;286
13.1;7.1 Introduction;286
13.2;7.2 Convergence Theorems;290
13.3;7.3 Martingales;298
13.4;7.4 Martingale Convergence Theorems;307
13.5;7.5 Uniform Integrability;312
13.6;7.6 Uniform Integrability and Martingale Theory;315
13.7;7.7 Optional Sampling Theorems;319
13.8;7.8 Applications of Martingale Theory;326
13.9;7.9 Applications to Markov Chains;333
13.10;7.10 References;337
14;Chapter 8. The Central Limit Theorem;338
14.1;8.1 Introduction;338
14.2;8.2 The Fundamental Weak Compactness Theorem;346
14.3;8.3 Convergence to a Normal Distribution;353
14.4;8.4 Stable Distributions;361
14.5;8.5 Infinitely Divisible Distributions;365
14.6;8.6 Uniform Convergence in the Central Limit Theorem;373
14.7;8.7 Proof of the Inversion Formula;376
14.8;8.8 Completion of the Proof of Theorem 8.3.2;378
14.9;8.9 Proof of the Convergence of Types Theorem (8.3.4);381
14.10;8.10 References;384
15;Appendix on General Topology;386
15.1;A1 Introduction;386
15.2;A2 Convergence;387
15.3;A3 Product and Quotient Topologies;393
15.4;A4 Separation Properties and Other Ways of Classifying Topological Spaces;396
15.5;A5 Compactness;398
15.6;A6 Semicontinuous Functions;405
15.7;A7 The Stone-Weierstrass Theorem;408
15.8;A8 Topologies on Function Spaces;411
15.9;A9 Complete Metric Spaces and Category Theorems;415
15.10;A10 Uniform Spaces;419
16;Bibliography;426
17;Solutions to Problems;428
18;Subject Index;486