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Probability Theory

Michael Loève
Dover Publications
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

Michel Loève (1907–1979) was a French-American mathematician who worked in probability theory and mathematical statistics. The present book was first published in 1955 and is a graduate-level text and a comprehensive handbook of probability as the subject existed at that time. The present volume is a 2017 Dover unaltered reprint of the 1963 Van Nostrand third edition.

This is an advanced text, aimed at graduate students and practitioners. It’s not for beginners, even though it starts at the beginning with a preliminary chapter on concepts of probability (all about discrete probability) and a comprehensive 100-page section on measure and integration. The pace is very fast, the writing is concise, and there are few examples. Each chapter ends with a “Complements and Details” section which contains the exercises and additional theorems that did not fit in the narrative. The book is skimpy on applications, and you almost never see a specific probability distribution. That being said, it does contain nearly anything you would ever want to know about the theory of probability.

Despite its age, the book holds up well. Two more-recent books with somewhat similar coverage, that are well-regarded but that I have not seen, are Kallenberg’s Foundations of Modern Probability (2nd edition, 2002) and Durrett’s Probability: Theory and Examples (4th edition, 2010).

This is not the latest edition: A fourth edition in two volumes was published in 1977–78 by Springer and is still in print. It is not drastically different from the present volume; it adds material on Brownian motion, functional limit distributions, and random walks.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.

Introductory Part: Elementary Probability Theory
I. Intuitive Background
II. Axioms; Independence and the Bernoulli Case
III. Dependence and Chains

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 \(\sigma\)-Fields

Chapter II: Measurable Functions and Integration
5. Measurable Functions; 6. Measure and Convergence; 7. Integration; 8. Indefinite Integrals, Iterated Integrals

Part Two: General Concepts and Tools of Probability Theory

Chapter III: Probability Concepts
9. Probability Spaces and Random Variables; 10. Probability Distributions

Chapter IV: Distribution Functions and Characteristic Functions
11. Distribution Functions; 12. Characteristic Functions and Distribution Functions; 13. Probability Laws and Types of Laws; 14. Nonnegative-definiteness, Regularity

Part Three: Independence

Chapter V: Sums of Independent Random Variables
15. Concept of Independence; 16. Convergence and Stability of Sums, Centering at Expectations and Truncation; 17. Convergence and Stability of Sums, Centering at Medians and Symmetrization; 18. Exponential Bounds and Normed Sums

Chapter VI: Central Limit Problem
19. Degenerate, Normal, and Poisson Types; 20. Evolution of the Problem; 21. Central Limit Problem, The Case of Bounded Variances; 22. Solution of the Central Limit Problem; 23. Normed Sums

Part Four: Dependence

Chapter VII: Conditioning
24. Concept of Conditioning; 25. Properties of Conditioning; 26. Regular Pr. Functions; 27. Conditional Distributions

Chapter VIII: From Independence to Dependence
28. Central Asymptotic Problem; 29. Centerings, Martingales, and A.S. Convergence

Chapter IX: Ergodic Theorems
30. Translation of Sequences, Basic Ergodic Theorem and Stationarity; 31. Ergodic Theorems and \(L_r\)-Spaces; 32. Ergodic Theorems on Banach Spaces

Chapter X: Second Order Properties
33. Orthogonality; 34. Second Order Random Functions

Part Five: Elements of Random Analysis

Chapter XI: Foundations, Martingales and Decomposability
35. Foundations; 36. Martingales; 37. Decomposability Chapter XII: Markov Processes 38. Markov Dependence; 39. Time-continuous Transition Probabilities; Markov Semi-groups; 41. Sample Continuity and Diffusion Operators