A Basic Course in Probability Theory

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

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