A modern approach to probability theory
著者
書誌事項
A modern approach to probability theory
（Probability and its applications）
Birkhäuser, c1997
- : us : hbk
- : sz : hbk
大学図書館所蔵 全48件
注記
Includes bibliographical references and index
内容説明・目次
- 巻冊次
-
: us : hbk ISBN 9780817638078
内容説明
Students and teachers of mathematics and related fields will find this book a comprehensive and modern approach to probability theory, providing the background and techniques to go from the beginning graduate level to the point of specialization in research areas of current interest. The book is designed for a two- or three-semester course, assuming only courses in undergraduate real analysis or rigorous advanced calculus, and some elementary linear algebra. A variety of applications-Bayesian statistics, financial mathematics, information theory, tomography, and signal processing-appear as threads to both enhance the understanding of the relevant mathematics and motivate students whose main interests are outside of pure areas.
目次
List of Tables * Preface * Part I: Probability Spaces, Random Variables, and Expectations * Probability Spaces * Random Variables * Distribution Functions * Expectations: Theory * Expectations: Applications * Calculating Probabilities and Measures * Measure Theory: Existence and Uniqueness * Integration Theory * Part 2: Independence and Sums * Stochastic Independence * Sums of Independent Random Variables * Random Walk * Theorems of A.S. Convergence * Characteristic Functions * Part 3: Convergence in Distribution * Convergence in Distribution on the Real Line * Distributional Limit Theorems for Partial Sums * Infinitely Divisible and Stable Distributions as Limits * Convergence in Distribution on Polish Spaces * The Invariance Principle and Brownian Motion * Part 4: Conditioning * Spaces of Random Variables * Conditional Probabilities * Construction of Random Sequences * Conditional Expectations * Part 5: Random Sequences * Martingales * Renewal Sequences * Time-homogeneous Markov Sequences * Exchangeable Sequences * Stationary Sequences * Part 6: Stochastic Processes * Point Processes * Diffusions and Stochastic Calculus * Applications of Stochastic Calculus * Part 7: Appendices * Appendix A. Notation and Usage of Terms * Appendix B. Metric Spaces * Appendix C. Topological Spaces * Appendix D. Riemann-Stieltjes Integration * Appendix E. Taylor Approximations, C-Valued Logarithms * Appendix F. Bibliography * Appendix G. Comments and Credits * Index
- 巻冊次
-
: sz : hbk ISBN 9783764338077
内容説明
This textbok is designed for graduate students in probability theory. It merges measure theory with probability theory, and rather than deal only with "random variables" it also looks at "random objects". There is a chapter of introductory material for advanced topics, as well as over 1000 exercises, ranging from gambling theory to concrete calculations involving random sets. All of the problems in the exercises are designed to help students get beyond mere rote learning of theorems and proofs to a deep intuitive feel for the far-reaching implications of the theory. Solutions are provided for approximately a quarter of the exercises.
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