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Probability theory is a rapidly expanding field and is used in many areas of science and technology. Beginning from a basis of abstract analysis, this mathematics book develops the knowledge needed for advanced students to develop a complex understanding of probability. The first part of the book systematically presents concepts and results from analysis before embarking on the study of probability theory. The initial section will also be useful for those interested in topology, measure theory, real analysis and functional analysis. The second part of the book presents the concepts, methodology and fundamental results of probability theory. Exercises are included throughout the text, not just at the end, to teach each concept fully as it is explained, including presentations of interesting extensions of the theory. The complete and detailed nature of the book makes it ideal as a reference book or for self-study in probability and related fields.
- Covers a wide range of subjects including f-expansions, Fuk-Nagaev inequalities and Markov triples.
- Provides multiple clearly worked exercises with complete proofs.
- Guides readers through examples so they can understand and write research papers independently.
Students and researchers in probability theory, topology, measure theory, real and functional analysis and related fields.
PART 1: ANALYSIS
Chapter 1. Elements of Set Theory
1 Sets and Operations on Sets
2 Functions and Cartesian Products
3 Equivalent Relations and Partial Orderings
Chapter 2. Topological Preliminaries
4 Construction of Some Topological Spaces
5 General Properties of Topological Spaces
6 Metric Spaces
Chapter 3. Measure Spaces
7 Measurable Spaces
8 Measurable Functions
9 Definitions and Properties of the Measure
10 Extending Certain Measures
Chapter 4. The Integral
11 Definitions and Properties of the Integral
12 Radon-Nikodým Theorem and the Lebesgue Decomposition
13 The Spaces
14 Convergence for Sequences of Measurable Functions
Chapter 5. Measures on Product σ-Algebras
15 The Product of a Finite Number of Measures
16 The Product of Infinitely Many Measures
PART 2: PROBABILITY
Chapter 6. Elementary Notions in Probability Theory
17 Events and Random Variables
18 Conditioning and Independence
Chapter 7. Distribution Functions and Characteristic Functions
19 Distribution Functions
20 Characteristic Functions
Chapter 8. Probabilities on Metric Spaces
21 Probabilities in a Metric Space
22 Topology in the Space of Probabilities
Chapter 9. Central Limit Problem
23 Infinitely Divisible Distribution/Characteristic Functions
24 Convergence to an Infinitely Divisible Distribution/Characteristic Function
Chapter 10. Sums of Independent Random Variables
25 Weak Laws of Large Numbers
26 Series of Independent Random Variables
27 Strong Laws of Large Numbers
28 Laws of the Iterated Logarithm
Chapter 11. Conditioning
29 Conditional Expectations, Conditional Probabilities and Conditional Independence
30 Stopping Times and Semimartingales
Chapter 12. Ergodicity, Mixing, and Stationarity
31 Ergodicity and Mixing
32 Stationary Sequences
List of Symbols
- No. of pages:
- © Elsevier 2013
- 11th January 2013
- Hardcover ISBN:
- eBook ISBN:
Institute of Mathematical Statistics and Applied Mathematics, Casa Academiei Romane, Bucharest, Romania
"...The subject of this book is modern probability theory and it gives a full self-contained presentation of fundamental results within the area…it presents the theory of probability from a measure-theoretic point of view."--MathSciNet, Analysis and Probability
"This monograph is a self-contained and compactly written text book on probability theory for graduate students…the textbook gives a rigorous…presentation of the fundamental results in probability theory with an emphasis on the interaction between analysis and probability."--Zentralblatt MATH, 1278.60002
"Spataru presents a probability theory text for graduate students who have a solid background in abstract analysis, which is required because he develops probability theory from a measure-theoretic perspective."--Reference & Research Book News, October 2013
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