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An Introduction to Measure-Theoretic Probability (2nd Ed.)

Langue : Anglais

Auteur :

Couverture de l’ouvrage An Introduction to Measure-Theoretic Probability

An Introduction to Measure-Theoretic Probability, Second Edition, employs a classical approach to teaching the basics of measure theoretic probability. 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.

This edition requires no prior knowledge of measure theory, covers all its topics in great detail, and includes one chapter on the basics of ergodic theory and one chapter on two cases of statistical estimation. Topics range from the basic properties of a measure to 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; standard moment and probability inequalities; the Hahn-Jordan Decomposition Theorem; the Lebesgue Decomposition T; conditional expectation and conditional probability; theory of characteristic functions; sequences of independent random variables; and ergodic theory. There is a considerable bend toward the way probability is actually used in statistical research, finance, and other academic and nonacademic applied pursuits. Extensive exercises and practical examples are included, and all proofs are presented in full detail. Complete and detailed solutions to all exercises are available to the instructors on the book companion site.

This text will be a valuable resource for graduate students primarily in statistics, mathematics, electrical and computer engineering or other information sciences, as well as for those in mathematical economics/finance in the departments of economics.

1. Certain Classes of Sets, Measurability, Pointwise Approximation2. Definition and Construction of a Measure and Its Basic Properties3. Some Modes of Convergence of a Sequence of Random Variables and Their Relationships4. The Integral of a Random Variable and Its Basic Properties5. Standard Convergence Theorems, The Fubini Theorem6. Standard Moment and Probability Inequalities, Convergence in the r-th Mean and Its Implications7. The Hahn-Jordan Decomposition Theorem, The Lebesgue Decomposition Theorem, and The Radon-Nikcodym Theorem8. Distribution Functions and Their Basic Properties, Helly-Bray Type Results9. Conditional Expectation and Conditional Probability, and Related Properties and Results10. Independence11. Topics from the Theory of Characteristic Functions12. The Central Limit Problem: The Centered Case13. The Central Limit Problem: The Noncentered Case14. Topics from Sequences of Independent Random Variables15. Topics from Ergodic Theory
Graduate students primarily in statistics, mathematics, electrical & computer engineering or other information sciences; mathematical economics/finance in departments of economics.
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.
  • Provides in a concise, yet detailed way, the bulk of probabilistic tools essential to a student working toward an advanced degree in statistics, probability, and other related fields
  • Includes extensive exercises and practical examples to make complex ideas of advanced probability accessible to graduate students in statistics, probability, and related fields
  • All proofs presented in full detail and complete and detailed solutions to all exercises are available to the instructors on book companion site
  • Considerable bend toward the way probability is used in statistics in non-mathematical settings in academic, research and corporate/finance pursuits

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Ouvrage de 426 p.

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116,98 €

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Mots-clés :

Additive constant; Additivity; Argument; Asymptotic; Auxiliary results; Bandwidth; Binomials; Borel real line; Borel-Cantelli Theorem; Cantor diagonal method; Cardinality; Cartesian product; Cauchy criterion; Central limit theorem; Complementation; Constant; Convergence theorem; Convex function; Convexity; Coordinate process; Correlation coefficient; Countable operations; Countable unions; Covariance; Covention; Decomposition; Delta method; Derivatives; Differentiation; Discontinuity points; Double exponential; Elementary definition; Ergodicity; Euclidean distance; Events; Extension; Factorization; Finite number; Finite variations; Finiteness; Fubini theorem; Fundamental Theorem; Grouping; Helly-Bray type; Indefinite integral; Independent classes; Indicator function; Induction hypothesis; Induction method; Inequalities; Integrands; Integration; Intersection reformulation; Invariance; Inversion formula; Jensen Inequality; Kernel method; Kolmogorov inequalities; Lebesgue measure; Lebesgue sense; Limiting distribution; Logarithm; Measurable mapping; Measurable sets; Mode; Moment; Monotone class; Mutual convergence; Noncentered case; Nonnegative function; Nonparametric estimation; Nonrestrictive assumption; Notation; Notational convenience; Null set; Open interval; Pairwise disjoint; Partial sums; Partition; Probability measure; Probability sense; Product space; Ramification; Ramifications; Random vector; Rationals; Regularity conditions; Reindexing; Riemann-Stieltjes sense; Robustness; Sample mean; Shift transformation; Signed measure; Singletons; Slutsky's theorem; Stationarity; Statistical inference; Step function; Stochastic; Subsequence; Subsets; Suffices; Summation; Taylor's theorem; Tchebichev inequality; Toeplitz Lemma; Triangular arrays; Trivial field; Union