An introduction to measure and probability

J.C. Taylor

Assuming only calculus and linear algebra, Professor Taylor introduces readers to measure theory and probability, discrete martingales, and weak convergence. This is a technically complete, self-contained and rigorous approach that helps the reader to develop basic skills in analysis and probability. Students of pure mathematics and statistics can thus expect to acquire a sound introduction to basic measure theory and probability, while readers with a background in finance, business, or engineering will gain a technical understanding of discrete martingales in the equivalent of one semester. J. C. Taylor is the author of numerous articles on potential theory, both probabilistic and analytic, and is particularly interested in the potential theory of symmetric spaces.

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[目次]

  • I. Probability Spaces.- 1. Introduction to ?.- 2. What is a probability space? Motivation.- 3. Definition of a probability space.- 4. Construction of a probability from a distribution function.- 5. Additional exercises*.- II. Integration.- 1. Integration on a probability space.- 2. Lebesgue measure on ? and Lebesgue integration.- 3. The Riemann integral and the Lebesgue integral.- 4. Probability density functions.- 5. Infinite series again.- 6. Differentiation under the integral sign.- 7. Signed measures and the Radon-Nikodym theorem*.- 8. Signed measures on ? and functions of bounded variation*.- 9. Additional exercises*.- III. Independence and Product Measures.- 1. Random vectors and Borel sets in ?n.- 2. Independence.- 3. Product measures.- 4. Infinite products.- 5. Some remarks on Markov chains*.- 6. Additional exercises*.- IV. Convergence of Random Variables and Measurable Functions.- 1. Norms for random variables and measurable functions.- 2. Continuous functions and Lp*.- 3. Pointwise convergence and convergence in measure or probability.- 4. Kolmogorov's inequality and the strong law of large numbers.- 5. Uniform integrability and truncation*.- 6. Differentiation: the Hardy-Littlewood maximal function*.- 7. Additional exercises*.- V. Conditional Expectation and an Introduction to Martingales.- 1. Conditional expectation and Hilbert space.- 2. Conditional expectation.- 3. Sufficient statistics*.- 4. Martingales.- 5. An introduction to martingale convergence.- 6. The three-series theorem and the Doob decomposition.- 7. The martingale convergence theorem.- VI. An Introduction to Weak Convergence.- 1. Motivation: empirical distributions.- 2. Weak convergence of probabilities: equivalent formulations.- 3. Weak convergence of random variables.- 4. Empirical distributions again: the Glivenko-Cantelli theorem.- 5. The characteristic function.- 6. Uniqueness and inversion of the characteristic function.- 7. The central limit theorem.- 8. Additional exercises*.- 9. Appendix*.

「Nielsen BookData」より

この本の情報

書名 An introduction to measure and probability
著作者等 Taylor, J. C.
Taylor J.C.
出版元 Springer
刊行年月 c1997
版表示 1st ed. 1997. Corr. 2nd printing 1998
ページ数 xvii, 299 p.
大きさ 24 cm
ISBN 0387948309
NCID BA29508339
※クリックでCiNii Booksを表示
言語 英語
出版国 アメリカ合衆国
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