Probability Theory I

This fourth edition contains several additions. The main ones con­ cern three closely related topics: Brownian motion, functional limit distributions, and random walks. Besides the power and ingenuity of their methods and the depth and beauty of their results, their importance is fast growing in Analysis as well as in theoretical and applied Proba­ bility. These additions increased the book to an unwieldy size and it had to be split into two volumes. About half of the first volume is devoted to an elementary introduc­ tion, then to mathematical foundations and basic probability concepts and tools. The second half is devoted to a detailed study of Independ­ ence which played and continues to playa central role both by itself and as a catalyst. The main additions consist of a section on convergence of probabilities on metric spaces and a chapter whose first section on domains of attrac­ tion completes the study of the Central limit problem, while the second one is devoted to random walks. About a third of the second volume is devoted to conditioning and properties of sequences of various types of dependence. The other two thirds are devoted to random functions; the last Part on Elements of random analysis is more sophisticated. The main addition consists of a chapter on Brownian motion and limit distributions.

of Volume I.- Introductory Part: Elementary Probability Theory.- I. Intuitive Background.- 1. Events.- 2. Random events and trials.- 3. Random variables.- II. Axioms; Independence and the Bernoulli Case.- 1. Axioms of the finite case.- 2. Simple random variables.- 3. Independence.- 4. Bernoulli case.- 5. Axioms for the countable case.- 6. Elementary random variables.- 7. Need for nonelementary random variables.- III. Dependence and Chains.- 1. Conditional probabilities.- 2. Asymptotically Bernoullian case.- 3. Recurrence.- 4. Chain dependence.- *5. Types of states and asymptotic behavior.- *6. Motion of the system.- *7. Stationary chains.- Complements and Details.- One: Notions of Measure Theory.- I: Sets, Spaces, and Measures.- 1. Sets, Classes, and Functions.- 1.1 Definitions and notations.- 1.2 Differences, unions, and intersections.- 1.3 Sequences and limits.- 1.4 Indicators of sets.- 1.5 Fields and ?-fields.- 1.6 Monotone classes.- *1.7 Product sets.- *1.8 Functions and inverse functions.- *1.9 Measurable spaces and functions.- *2. Topological Spaces.- *2.1 Topologies and limits.- *2.2 Limit points and compact spaces.- *2.3 Countability and metric spaces.- *2.4 Linearity and normed spaces.- 3. Additive Set Functions.- 3.1 Additivity and continuity.- 3.2 Decomposition of additive set functions.- *4. Construction of Measures on ?-Fields.- *4.1 Extension of measures.- *4.2 Product probabilities.- *4.3 Consistent probabilities on Borel fields.- *4.4 Lebesgue-Stieltjes measures and distribution functions.- Complements and Details.- II: Measurable Functions and Integration.- 5. Measurable Functions.- 5.1 Numbers.- 5.2 Numerical functions.- 5.3 Measurable functions.- 6. Measure and Convergences.- 6.1 Definitions and general properties.- 6.2 Convergence almost everywhere.- 6.3 Convergence in measure.- 7. Integration.- 7.1 Integrals.- 7.2 Convergence theorems.- 8. Indefinite Integrals; Iterated Integrals.- 8.1 Indefinite integrals and Lebesgue decomposition.- 8.2 Product measures and iterated integrals.- *8.3 Iterated integrals and infinite product spaces.- Complements and Details.- Two: General Concepts and Tools of Probability Theory.- III: Probability Concepts.- 9. Probability Spaces and Random Variables.- 9.1 Probability terminology.- *9.2 Random vectors, sequences, and functions.- 9.3 Moments, inequalities, and convergences.- *9.4 Spaces L?.- 10. Probability Distributions.- 10.1 Distributions and distribution functions.- 10.2 The essential feature of pr. theory.- Complements and Details.- IV: Distribution Functions and Characteristic Functions.- 11. Distribution Functions.- 11.1 Decomposition.- 11.2 Convergence of d.f.’s.- 11.3 Convergence of sequences of integrals.- *11.4 Further extension and convergence of moments..- *11.5 Discussion.- *12. Convergence of Probabilities on Metric Spaces.- *12.1 Convergence.- *12.2 Regularity and tightness.- *12.3 Tightness and relative compactness.- 13. Characteristic Functions and Distribution Functions.- 13.1 Uniqueness.- 13.2 Convergences.- 13.3 Composition of d.f.’s and multiplication of ch.f.’s..- 13.4 Elementary properties of ch.f.’s and first applications.- 14. Probability Laws and Types of Laws.- 14.1 Laws and types; the degenerate type.- 14.2 Convergence of types.- 14.3 Extensions.- 15. Nonnegative-definiteness; Regularity.- 15.1 Ch.f.’s and nonnegative-definiteness.- *15.2 Regularity and extension of ch.f.’s.- *15.3 Composition and decomposition of regular ch.f.’s..- Complements and Details.- Three: Independence.- V: Sums of Independent Random Variables.- 16. Concept of Independence.- 16.1 Independent classes and independent functions.- 16.2 Multiplication properties.- 16.3 Sequences of independent r.v.’s.- *16.4 Independent r.v.’s and product spaces.- 17. Convergence and Stability of Sums; Centering at Expectations and Truncation.- 17.1 Centering at expectations and truncation.- 17.2 Bounds in terms of variances.- 17.3 Convergence and stability.- *17.4 Generalization.- *18. Convergence and Stability of Sums; Centering at Medians and Symmetrization.- *18.1 Centering at medians and symmetrization.- *18.2 Convergence and stability.- *19. Exponential Bounds and Normed Sums.- *19.1 Exponential bounds.- *19.2 Stability.- *19.3 Law of the iterated logarithm.- Complements and Details.- VI: Central Limit Problem.- 20. Degenerate, Normal, and Poisson Types.- 20.1 First limit theorems and limit laws.- *20.2 Composition and decomposition.- 21. Evolution of the Problem.- 21.1 The problem and preliminary solutions.- 21.2 Solution of the Classical Limit Problem.- *21.3 Normal approximation.- 22. Central Limit Problem; the Case of Bounded Variances.- 22.1 Evolution of the problem.- 22.2 The case of bounded variances.- *23. Solution of the Central Limit Problem.- *23.1 A family of limit laws; the infinitely decomposable laws.- *23.2 The uan condition.- *23.3 Central Limit Theorem.- *23.4 Central convergence criterion.- 23.5 Normal, Poisson, and degenerate convergence..- *24. Normed Sums.- *24.1 The problem.- *24.2 Norming sequences.- *24.3 Characterization of 31.- *24.4 Identically distributed summands and stable laws..- 24.5 Lévy representation.- Complements and Details.- VII: Independent Identically Distributed Summands.- 25. Regular Variation and Domains of Attraction.- 25.1 Regular variation.- 25.2 Domains of attraction.- 26. Random Walk.- 26.1 Set-up and basic implications.- 26.2 Dichotomy: recurrence and transience.- 26.3 Fluctuations; exponential identities.- 26.4 Fluctuations; asymptotic behaviour.- Complements and Details.