This fourth edition contains several additions. The main ones con­cern 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 Proba­bility. 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 introduction, then to mathematical foundations and basic probability concepts and tools. The second half is devoted to a detailed study of Independence which played and continues to play a 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 attraction 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. 

<Contents 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

Part One: Notions of Measure Theory

Chapter I: Sets, Spaces, and Measures

1. Sets, Classes, and Functions

*2. Topological Spaces

3. Additive Set Functions

*4. Construction of Measures on σ-Fields

Complements and Details

Chapter II: Measurable Functions and Integration

5. Measurable Functions

6. Measure and Convergences

7. Integration

8. Indefinite Integrals; Iterated Integrals

Complements and Details

Part Two: General Concepts and Tools of Probability Theory

Chapter III: Probability Concepts

9. Probability Spaces and Random Variables

10. Probability Distributions

Complements and Details

Chapter IV: Distribution Functions and Characteristic Functions

11. Distribution Functions

*12. Convergence of Probabilities on Metric Spaces

13. Characteristic Functions and Distribution Functions

14. Probability Laws and Types of Laws

15. Nonnegative-definiteness; Regularity

Complements and Details

Part Three: Independence

Chapter V: Sums of Independent Random Variables

16. Concept of Independence

17. Convergence and Stability of Sums; Centering at Expectations and Truncation

*18. Convergence and Stability of Sums; Centering at Medians and Symmetrization

*19. Exponential Bounds and Normed Sums

Complements and Details

Chapter VI: Central Limit Problem

20. Degenerate, Normal, and Poisson Types

21. Evolution of The Problem

22. Central Limit Problem; The Case of Bounded Variances

*23. Solution of The Central Limit Problem

*24. Normed Sums

Complements and Details

Chapter VII: Independent Identically Distributed Summands

25. Regular Variation and Domains of Attraction

26. Random Walk

Complements and Details

Bibliography

Index

<Contents of Volume II>

PART FOUR: Dependence

Chapter VIII: Conditioning

27. Concept of Conditioning

28. Properties of Conditioning

29. Regular Pr. Functions

30. Conditional Distributions

Complements and Details

Chapter IX: From Independence to Dependence

31. Central Asymptotic Problem

32. Centerings, Martingales, and a.s. Convergence

Complements and Details

Chapter X: Ergodic Theorems

33. Translation of Sequences; Basic Ergodic Theorem and Stationarity

*34. Ergodic Theorems and Lr-Spaces

*35. Ergodic Theorems on Banach Spaces

Complements and Details

Chapter XI: Second Order Properties

36. Orthogonality

37. Second Order Random Functions

Complements and Details

PART FIVE: Elements of Random Analysis

Chapter XII: Foundations; Martingales and Decomposability

38. Foundations

39. Martingales

40. Decomposability

Complements and Details

Chapter XIII: Brownian Motion and Limit Distributions

41. Brownian Motion

42. Limit Distributions

Complements and Details

Chapter XIV: Markov Processes

43. Markov Dependence

44. Time-Continuous Transition Probabilities

45. Markov Semi-Groups

46. Sample Continuity and Diffusion Operators

Complements and Details

Bibliography

Index

지은이: Michel Loeve

Departments of Mathematics and Statistics University of California at Berkeley

Michel Loeve is Professor Emeritus of mathematics and statistics at the University of California at Berkeley. Educated in France, he has taught in the United States for many years and held visiting professorships at Columbia University and the Universite de Paris. Dr. Loeve's interest is directed towards problems in pure and applied probability theory, a fact which is also reflected in the wide and encyclopaedic character of this book.

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