Bert E. Fristedt, Lawrence F. Gray

A Modern Approach to Probability Theory (eBook, PDF)

• Format: PDF

Produktbeschreibung

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.

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Produktdetails

• Verlag: Birkhäuser Boston
• Seitenzahl: 758
• Erscheinungstermin: 21. November 2013
• Englisch
• ISBN-13: 9781489928375
• Artikelnr.: 44195098

Inhaltsangabe

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

Rezensionen

"Covers the essentials in a clear and readable fashion... A must for professionals and an attractive text for a graduate course." -- American Mathematical Monthly (review of the first edition)

"This ambitious book is intended as 'a textbook in probability for graduate students in mathematics and related areas such as economics, statistics, physics and operations research'...The coverage is careful and thorough...Quite a lot of fairly recent material is incorporated, and this is certainly one of the book's strengths. The selection of material is sensible, and the quality of exposition is good...In sum: the book contains a lot of good mathematics, nicely done, and should prove useful to students and teachers, and to specialists in probability theory." -- Mathematical Reviews (review of the first edition)

"The book takes the reader from a relatively low level.... To the point where he or she can specialize in research topics of current interest. ...an outstanding basis for teaching a graduate course in probability theory. The exhaustive compilation of results and detailed index also make it a very useful reference text for the more advanced probabilist... a good buy for anyone looking for a very accessible and complete mathematical account of modern probability theory." -- Journal of the American Statistical Association (review of the first edition)