## A Modern Approach to Probability TheoryOverview 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. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

3 | |

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 |

Lévy 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 |

715 | |

Appendix G Comments and Credits | 723 |

### Other editions - View all

### Common terms and phrases

a-field Borel space bounded calculate Chapter characteristic function compact conditional distribution constant convergence in distribution Convergence Theorem Corollary corresponding countable definition denote density distribution function distribution Q domain equal ergodic Example exchangeable sequence exists filtration Finetti measure formula Fubini Theorem function F given Hint iid sequence independent inequality interval Jºn Large Numbers Lebesgue measure Lemma Let Q Lévy measure Lévy process lim sup limit Markov sequence martingale measurable function measurable space metric space nonnegative obtain Poisson point process Polish space positive integer preceding problem preceding proposition preceding theorem probability generating function probability measure probability space Problem 13 Problem 9 Prove the preceding R-valued random variables R(ds random sequence random walk real number renewal sequence result Riemann-Stieltjes integral space Q stationary sequence Suppose topology transition operator values variance