• 1. Measure theory--
• 2. Laws of large numbers--
• 3. Central limit theorems--
• 4. Random walks--
• 5. Martingales--
• 6. Markov chains--
• 7. Ergodic theorems--
• 8. Brownian motion-- Appendix A. Measure theory details.
• (source: Nielsen Book Data)

This classic introduction to probability theory for beginning graduate students covers laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems. The new edition begins with a short chapter on measure theory to orient readers new to the subject.
(source: Nielsen Book Data)

• 1. Foundations: set theory--
• 2. General topology--
• 3. Measures--
• 4. Integration--
• 5. Lp spaces: introduction to functional analysis--
• 6. Convex sets and duality of normed spaces--
• 7. Measure, topology, and differentiation--
• 8. Introduction to probability theory--
• 9. Convergence of laws and central limit theorems--
• 10. Conditional expectations and martingales--
• 11. Convergence of laws on separable metric spaces--
• 12. Stochastic processes--
• 13. Measurability: Borel isomorphism and analytic sets-- Appendixes: A. Axiomatic set theory-- B. Complex numbers, vector spaces, and Taylor's theorem with remainder-- C. The problem of measure-- D. Rearranging sums of nonnegative terms-- E. Pathologies of compact nonmetric spaces-- Indices.
• (source: Nielsen Book Data)

This classic textbook offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The first half of the book gives an exposition of real analysis: basic set theory, general topology, measure theory, integration, an introduction to functional analysis in Banach and Hilbert spaces, convex sets and functions and measure on topological spaces. The second half introduces probability based on measure theory, including laws of large numbers, ergodic theorems, the central limit theorem, conditional expectations and martingale's convergence. A chapter on stochastic processes introduces Brownian motion and the Brownian bridge. The edition has been made even more self-contained than before; it now includes a foundation of the real number system and the Stone-Weierstrass theorem on uniform approximation in algebras of functions. Several other sections have been revised and improved, and the comprehensive historical notes have been further amplified. A number of new exercises have been added, together with hints for solution.
(source: Nielsen Book Data)

• 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-- Appendices-- Exercises.
• (source: Nielsen Book Data)

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 role. 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.
(source: Nielsen Book Data)