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Analysis and Probability

Book 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.

  • 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.

Table of Contents

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright
  5. Preface
    1. Chapter 1. Elements of Set Theory
      1. 1 Sets and Operations on Sets
      2. 2 Functions and Cartesian Products
      3. 3 Equivalent Relations and Partial Orderings
      4. References
    2. Chapter 2. Topological Preliminaries
      1. 4 Construction of Some Topological Spaces
      2. 5 General Properties of Topological Spaces
      3. 6 Metric Spaces
    3. Chapter 3. Measure Spaces
      1. 7 Measurable Spaces
      2. 8 Measurable Functions
      3. 9 Definitions and Properties of the Measure
      4. 10 Extending Certain Measures
    4. Chapter 4. The Integral
      1. 11 Definitions and Properties of the Integral
      2. 12 Radon-Nikodým Theorem and the Lebesgue Decomposition
      3. 13 The Spaces
      4. 14 Convergence for Sequences of Measurable Functions
    5. Chapter 5. Measures on Product σ-Algebras
      1. 15 The Product of a Finite Number of Measures
      2. 16 The Product of Infinitely Many Measures
    1. Chapter 6. Elementary Notions in Probability Theory
      1. 17 Events and Random Variables
      2. 18 Conditioning and Independence
    2. Chapter 7. Distribution Functions and Characteristic Functions
      1. 19 Distribution Functions
      2. 20 Characteristic Functions
      3. References
    3. Chapter 8. Probabilities on Metric Spaces
      1. 21 Probabilities in a Metric Space
      2. 22 Topology in the Space of Probabilities
    4. Chapter 9. Central Limit Problem
      1. 23 Infinitely Divisible Distribution/Characteristic Functions
      2. 24 Convergence to an Infinitely Divisible Distribution/Characteristic Function
      3. Reference
    5. Chapter 10. Sums of Independent Random Variables
      1. 25 Weak Laws of Large Numbers
      2. 26 Series of Independent Random Variables
      3. 27 Strong Laws of Large Numbers
      4. 28 Laws of the Iterated Logarithm
    6. Chapter 11. Conditioning
      1. 29 Conditional Expectations, Conditional Probabilities and Conditional Independence
      2. 30 Stopping Times and Semimartingales
    7. Chapter 12. Ergodicity, Mixing, and Stationarity
      1. 31 Ergodicity and Mixing
      2. 32 Stationary Sequences
  8. List of Symbols