## Key Features

- 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.

## Description

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.

Readership

Students and researchers in probability theory, topology, measure theory, real and functional analysis and related fields.

Analysis and Probability, 1st Edition

Preface

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

References

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

References

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

Reference

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