## A Basic Course in Probability TheoryIntroductory Probability is a pleasure to read and provides a fine answer to the question: How do you construct Brownian motion from scratch, given that you are a competent analyst? There are at least two ways to develop probability theory. The more familiar path is to treat it as its own discipline, and work from intuitive examples such as coin flips and conundrums such as the Monty Hall problem. An alternative is to first develop measure theory and analysis, and then add interpretation. Bhattacharya and Waymire take the second path. To illustrate the authors' frame of reference, consider the two definitions they give of conditional expectation. The first is as a projection of L2 spaces. The authors rely on the reader to be familiar with Hilbert space operators and at a glance, the connection to probability may not be not apparent. Subsequently, there is a discusssion of Bayes's rule and other relevant probabilistic concepts that lead to a definition of conditional expectation as an adjustment of random outcomes from a finer to a coarser information set. |

### What people are saying - Write a review

### Contents

1 | |

Exercises | 14 |

Independence Conditional Expectation | 19 |

Exercises | 33 |

Martingales and Stopping Times | 37 |

Exercises | 47 |

Classical ZeroOne Laws Laws of Large Numbers and Deviations | 49 |

Exercises | 57 |

Kolmogorovs Extension Theorem and Brownian Motion | 129 |

Exercises | 138 |

Brownian Motion The LIL and Some FineScale Properties | 141 |

Exercises | 145 |

Skorokhod Embedding and Donskers Invariance Principle | 147 |

Exercises | 164 |

A Historical Note on Brownian Motion | 166 |

Measure and Integration | 171 |

Work Convergence of Probability Measures | 59 |

Exercises | 70 |

Fourier Series Fourier Transform and Characteristics Functions | 73 |

Exercises | 93 |

Classical Central Limit Theorems | 99 |

Exercises | 104 |

Laplace Transforms and Tauberian Theorem | 106 |

Exercises | 118 |

Random Series of Independent Summands | 121 |

Exercises | 127 |

2 Integration and Basic Convergence Theorems | 176 |

3 Product Measures | 182 |

4 Riesz Representation on CS | 183 |

Topology and Function Spaces | 187 |

Hilbert Spaces and Applications in Measure Theory | 193 |

2 Lebegue Decomposition and the RadonNikodym Theorem | 196 |

200 | |

205 | |

211 | |