An Introduction to Measure-theoretic Probability

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Gulf Professional Publishing, 2005 - Mathematics - 443 pages
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This book provides in a concise, yet detailed way, the bulk of the probabilistic tools that a student working toward an advanced degree in statistics,
probability and other related areas, should be equipped with. The approach is classical, avoiding the use of mathematical tools not necessary for carrying out the discussions. All proofs are presented in full detail.

* Excellent exposition marked by a clear, coherent and logical devleopment of the subject
* Easy to understand, detailed discussion of material
* Complete proofs
 

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Contents

CHAPTER 1 Certain Classes of Sets Measurability and Pointwise Approximation
1
CHAPTER 2 Definition and Construction of a Measure and Its Basic Properties
29
CHAPTER 3 Some Modes of Convergence of Sequences of Random Variables and Their Relationships
55
CHAPTER 4 The Integral of a Random Variable and Its Basic Properties
71
Standard Convergence Theorems the Fubini Theorem
89
Standard Moment and Probability Inequalities Convergence in the rth Mean and Its Implications
119
CHAPTER 7 The HahnJordan Decomposition Theorem the Lebesgue Decomposition Theorem and the RadonNikodym Theorem
147
CHAPTER 8 Distribution Functions and Their Basic Properties HellyBray Type Results
167
CHAPTER 11 Topics from the Theory of Characteristic Functions
235
The Centered Case
289
The Central Limit Problem The Noncentered Case
325
Topics from Sequences of Independent Random Variables
345
Topics from Ergodic Theory
383
Appendix
421
Selected References
431
Index
433

CHAPTER 9 Conditional Expectation and Conditional Probability and Related Properties and Results
187
CHAPTER 10 Independence
217

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About the author (2005)

George G. Roussas earned a B.S. in Mathematics with honors from the University of Athens, Greece, and a Ph.D. in Statistics from the University of California, Berkeley. As of July 2014, he is a Distinguished Professor Emeritus of Statistics at the University of California, Davis. Roussas is the author of five books, the author or co-author of five special volumes, and the author or co-author of dozens of research articles published in leading journals and special volumes. He is a Fellow of the following professional societies: The American Statistical Association (ASA), the Institute of Mathematical Statistics (IMS), The Royal Statistical Society (RSS), the American Association for the Advancement of Science (AAAS), and an Elected Member of the International Statistical Institute (ISI); also, he is a Corresponding Member of the Academy of Athens. Roussas was an associate editor of four journals since their inception, and is now a member of the Editorial Board of the journal Statistical Inference for Stochastic Processes. Throughout his career, Roussas served as Dean, Vice President for Academic Affairs, and Chancellor at two universities; also, he served as an Associate Dean at UC-Davis, helping to transform that institution's statistical unit into one of national and international renown. Roussas has been honored with a Festschrift, and he has given featured interviews for the Statistical Science and the Statistical Periscope. He has contributed an obituary to the IMS Bulletin for Professor-Academician David Blackwell of UC-Berkeley, and has been the coordinating editor of an extensive article of contributions for Professor Blackwell, which was published in the Notices of the American Mathematical Society and the Celebratio Mathematica.

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