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by R.M Gray

TOC

1 Probability and Random Processes
1.1 Introduction
1.2 Probability Spaces and Random Variables
1.3 Random Processes and Dynamical Systems
1.4 Distributions
1.5 Extension
1.6 Isomorphism

2 Standard alphabets
2.1 Extension of Probability Measures
2.2 Standard Spaces
2.3 Some properties of standard spaces
2.4 Simple standard spaces
2.5 Metric Spaces
2.6 Extension in Standard Spaces
2.7 The Kolmogorov Extension Theorem
2.8 Extension Without a Basis

3 Borel Spaces and Polish alphabets
3.1 Borel Spaces
3.2 Polish Spaces
3.3 Polish Schemes

4 Averages
4.1 Introduction
4.2 Discrete Measurements
4.3 Quantization
4.4 Expectation
4.5 Time Averages
4.6 Convergence of Random Variables
4.7 Stationary Averages

5 Conditional Probability and Expectation
5.1 Introduction
5.2 Measurements and Events
5.3 Restrictions of Measures
5.4 Elementary Conditional Probability
5.5 Projections
5.6 The Radon-Nikodym Theorem
5.7 Conditional Probability
5.8 Regular Conditional Probability
5.9 Conditional Expectation
5.10 Independence and Markov Chains

6 Ergodic Properties
6.1 Ergodic Properties of Dynamical Systems
6.2 Some Implications of Ergodic Properties
6.3 Asymptotically Mean Stationary Processes
6.4 Recurrence
6.5 Asymptotic Mean Expectations
6.6 Limiting Sample Averages
6.7 Ergodicity

7 Ergodic Theorems
7.1 Introduction
7.2 The Pointwise Ergodic Theorem
7.3 Block AMS Processes
7.4 The Ergodic Decomposition
7.5 The Subadditive Ergodic Theorem

8 Process Metrics and the Ergodic Decomposition
8.1 Introduction
8.2 A Metric Space of Measures
8.3 The Rho-Bar Distance
8.4 Measures on Measures
8.5 The Ergodic Decomposition Revisited
8.6 The Ergodic Decomposition of Markov Processes
8.7 Barycenters
8.8 Affine Functions of Measures
8.9 The Ergodic Decomposition of Affine Functionals

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