A Modern Approach to Probability Theory

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Springer Science & Business Media, Nov 21, 2013 - Mathematics - 758 pages
Overview This book is intended as a textbook in probability for graduate students in math ematics and related areas such as statistics, economics, physics, and operations research. Probability theory is a 'difficult' but productive marriage of mathemat ical abstraction and everyday intuition, and we have attempted to exhibit this fact. Thus we may appear at times to be obsessively careful in our presentation of the material, but our experience has shown that many students find them selves quite handicapped because they have never properly come to grips with the subtleties of the definitions and mathematical structures that form the foun dation of the field. Also, students may find many of the examples and problems to be computationally challenging, but it is our belief that one of the fascinat ing aspects of prob ability theory is its ability to say something concrete about the world around us, and we have done our best to coax the student into doing explicit calculations, often in the context of apparently elementary models. The practical applications of probability theory to various scientific fields are far-reaching, and a specialized treatment would be required to do justice to the interrelations between prob ability and any one of these areas. However, to give the reader a taste of the possibilities, we have included some examples, particularly from the field of statistics, such as order statistics, Dirichlet distri butions, and minimum variance unbiased estimation.
 

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Contents

Probability Spaces
3
Random Variables
11
Distribution Functions
25
Theory
41
Applications
59
Calculating Probabilities and Measures
75
Existence and Uniqueness
85
Integration Theory
101
Conditional Probabilities
403
Construction of Random Sequences
429
Conditional Expectations
443
Martingales
459
Renewal Sequences
489
Timehomogeneous Markov Sequences
511
Exchangeable Sequences
533
Stationary Sequences
553

Stochastic Independence
121
Sums of Independent Random Variables
147
Random Walk
163
Theorems of A S Convergence
185
Characteristic Functions
209
Convergence in Distribution on the Real Line
243
Distributional Limit Theorems for Partial Sums
271
Infinitely Divisible Distributions as Limits
289
Stable Distributions as Limits
323
Convergence in Distribution on Polish Spaces
347
The Invariance Principle and Brownian Motion
367
Spaces of Random Variables
395
Point Processes
581
LÚvy Processes
601
Introduction to Markov Processes
621
Interacting particle systems
641
Diffusions and Stochastic Calculus
661
Appendix A Notation and Usage of Terms
687
Appendix B Metric Spaces
693
RiemannStieltjes Integration
703
Appendix E Taylor Approximations CValued Logarithms
709
Appendix F Bibliography
715
Appendix G Comments and Credits
723
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