# A Basic Course in Probability Theory

Springer Science & Business Media, Jul 8, 2007 - Mathematics - 220 pages

Introductory 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.

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### Contents

 Random Maps Distribution and Mathematical Expectation 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 References 200 Index 205 Symbol Index 211 Copyright

### Popular passages

Page 202 - Teugels (1987): Regular Variation, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge.