A Modern Approach to Probability Theory

1. Front Matter

   Pages i-xx

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2. Probability Spaces, Random Variables, and Expectations

   1. Front Matter

      Pages 1-2

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   2. Probability Spaces

      • Bert Fristedt, Lawrence Gray

      Pages 3-10

   3. Random Variables

      • Bert Fristedt, Lawrence Gray

      Pages 11-24

   4. Distribution Functions

      • Bert Fristedt, Lawrence Gray

      Pages 25-40

   5. Expectations: Theory

      • Bert Fristedt, Lawrence Gray

      Pages 41-58

   6. Expectations: Applications

      • Bert Fristedt, Lawrence Gray

      Pages 59-74

   7. Calculating Probabilities and Measures

      • Bert Fristedt, Lawrence Gray

      Pages 75-84

   8. Measure Theory: Existence and Uniqueness

      • Bert Fristedt, Lawrence Gray

      Pages 85-100

   9. Integration Theory

      • Bert Fristedt, Lawrence Gray

      Pages 101-118

3. Independence and Sums

   1. Front Matter

      Pages 119-120

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   2. Stochastic Independence

      • Bert Fristedt, Lawrence Gray

      Pages 121-146

   3. Sums of Independent Random Variables

      • Bert Fristedt, Lawrence Gray

      Pages 147-162

   4. Random Walk

      • Bert Fristedt, Lawrence Gray

      Pages 163-184

   5. Theorems of A.S. Convergence

      • Bert Fristedt, Lawrence Gray

      Pages 185-208

   6. Characteristic Functions

      • Bert Fristedt, Lawrence Gray

      Pages 209-240

4. Convergence in Distribution

   1. Front Matter

      Pages 241-242

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   2. Convergence in Distribution on the Real Line

      • Bert Fristedt, Lawrence Gray

      Pages 243-270

   3. Distributional Limit Theorems for Partial Sums

      • Bert Fristedt, Lawrence Gray

      Pages 271-288

   4. Infinitely Divisible Distributions as Limits

      • Bert Fristedt, Lawrence Gray

      Pages 289-322

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.

• Algebra
• Finite
• Probability space
• Probability theory
• Random variable
• Sage
• Variable
• Variance
• bayesian statistics
• calculus
• function
• mathematics
• measure theory
• theorem

"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)

• School of Mathematics, University of Minnesota, Minneapolis, USA

  Bert Fristedt, Lawrence Gray

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