Probability with Martingales

Cambridge University Press, 14 de fev. de 1991
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.

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Preface please read
MeasureSpaces 1 0 Introductory remarks 1 1 Definitions of algebra σalgebra
Events 2 1 Model forexperiment Ω
Random Variables
An Easy StrongLaw 7 1 Independence means multiply again 7 2 Strong Lawfirstversion
Conditional Expectation
Martingales 10 1 Filtered spaces
Basic Properties of
Weak Convergence
The Central Limit Theorem
Appendix to Chapter 1
Appendix toChapter 5

The Convergence Theorem 11 1 The picture thatsays itall 11 2 Upcrossings 11 3 Doobs Upcrossing Lemma 11 4 Corollary
AStrong Law undervariance constraints
UniformIntegrability 13 1 An absolute continuity property 13 2 Definition UIfamily 13 3 Two simplesufficientconditions fortheUI property
Doobs OptionalSampling Theorem for UI martingales
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