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
Graduate students primarily in statistics, mathematics, electrical & computer engineering or other information sciences; mathematical economics/finance in departments of economics.
- Certain Classes of Sets, Measurability, Pointwise Approximation
- Definition and Construction of a Measure and Its Basic Properties
- Some Modes of Convergence of a Sequence of Random Variables and Their Relationships
- The Integral of a Random Variable and Its Basic Properties
- Standard Convergence Theorems, The Fubini Theorem
- Standard Moment and Probability Inequalities, Convergence in the r-th Mean and Its Implications
- The Hahn-Jordan Decomposition Theorem, The Lebesgue Decomposition Theorem, and The Radon-Nikcodym Theorem
- Distribution Functions and Their Basic Properties, Helly-Bray Type Results
- Conditional Expectation and Conditional Probability, and Related Properties and Results
- Topics from the Theory of Characteristic Functions
- The Central Limit Problem: The Centered Case 13. The Central Limit Problem: The Noncentered Case
- Topics from Sequences of Independent Random Variables
- Topics from Ergodic Theory
- Academic Press
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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.
University of California, Davis, USA
"...provides basic tools in measure theory and probability, in the classical spirit, relying heavily on characteristic functions as tools without using martingale or empirical process methods. A well-written book. Highly recommended [for] graduate students; faculty." -CHOICE "Based on the material presented in the manuscript, I would without any hesitation adopt the published version of the book. The topics dealt are essential to the understanding of more advanced material; the discussion is deep and it is combined with the use of essential technical details. It will be an extremely useful book. In addition it will be a very popular book." - Madan Puri, Indiana University "Would likely use as one of two required references when I teach either Stat 709 or Stat 732 again. Would also highly recommend to colleagues. The author has written other excellent graduate texts in mathematical statistics and contiguity and this promises to be another. This book could well become an important reference for mathematical statisticians. - Richard Johnson, University of Wisconsin "The author has succeeded in making certain deep and fundamental ideas of probability and measure theory accessible to statistics majors heading in the direction of graduate studies in statistical theory. " -Doraiswamy Ramachandran, California State University