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Probability with Martingales

Probability with Martingales


  • Author: David Williams, Statistical Laboratory, University of Cambridge
  • Date Published: February 1991
  • availability: Available
  • format: Paperback
  • isbn: 9780521406055

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About the Authors
  • Probability theory is nowadays applied in a huge variety of fields including physics, engineering, biology, economics and the social sciences. This book is a modern, lively and rigorous account which has Doob's theory of martingales in discrete time as its main theme. It proves important results such as Kolmogorov's Strong Law of Large Numbers and the Three-Series Theorem by martingale techniques, and the Central Limit Theorem via the use of characteristic functions. A distinguishing feature is its determination to keep the probability flowing at a nice tempo. It achieves this by being selective rather than encyclopaedic, presenting only what is essential to understand the fundamentals; and it assumes certain key results from measure theory in the main text. These measure-theoretic results are proved in full in appendices, so that the book is completely self-contained. The book is written for students, not for researchers, and has evolved through several years of class testing. Exercises play a vital rôle. Interesting and challenging problems, some with hints, consolidate what has already been learnt, and provide motivation to discover more of the subject than can be covered in a single introduction.

    • A modern, lively and rigorous account of probability theory using discrete-time martingales as the main theme
    • The treatment is selective, not encyclopaedic, and presents the essentials in a class-tested manner suitable for students
    • Interesting and challenging exercises consolidate what has already been learnt, and provide motivation to investigate the subject further
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    Reviews & endorsements

    '… one of the best introductions to Martingale theory.' Monatshefte für Mathematik

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    Product details

    • Date Published: February 1991
    • format: Paperback
    • isbn: 9780521406055
    • length: 265 pages
    • dimensions: 228 x 152 x 17 mm
    • weight: 0.412kg
    • contains: 3 b/w illus.
    • availability: Available
  • Table of Contents

    1. A branching-process example
    Part I. Foundations:
    2. Measure spaces
    3. Events
    4. Random variables
    5. Independence
    6. Integration
    7. Expectation
    8. An easy strong law: product measure
    Part II. Martingale Theory:
    9. Conditional expectation
    10. Martingales
    11. The convergence theorem
    12. Martingales bounded in L2
    13. Uniform integrability
    14. UI martingales
    15. Applications
    Part III. Characteristic Functions:
    16. Basic properties of CFs
    17. Weak convergence
    18. The central limit theorem

  • Instructors have used or reviewed this title for the following courses

    • Foundations of Probability and Statistics
    • Measure Theory for Probability
    • Probability and Stats I
  • Author

    David Williams, Statistical Laboratory, University of Cambridge

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