Probability for Statisticians

Preface<div>Use of This Text</div><div>Definition of Symbols</div><div><br></div><div>Chapter 1. Measures</div><div><ol><li>Basic Properties of Measures</li><li>Construction and Extension of Measures</li><li>Lebesgue Stieltjes Measures</li></ol></div>Chapter 2. Measurable Functions and Convergence<div><ol><li>Mappings and&nbsp;σ-Fields</li><li>Measurable Functions</li><li>Convergence</li><li>Probability, RVs, and Convergence in Law</li><li>Discussion of Sub&nbsp;σ-Fields</li></ol></div><div>Chapter 3. Integration</div><div><ol><li>The Lebesgue Integral</li><li>Fundamental Properties of Integrals</li><li>Evaluating and Differentiating Integrals</li><li>Inequalities</li><li>Modes of Convergence</li></ol></div><div>Chapter 4 Derivatives via Signed Measures</div><div><ol><li>Introduction</li><li>Decomposition of Signed Measures</li><li>The Radon Nikodym Theorem</li><li>Lebesgue's Theorem</li><li>The Fundamental Theorem of Calculus</li></ol></div><div>Chapter 5. Measures and Processes on Products</div><div><ol><li>Finite-Dimensional Product Spaces</li><li>Random Vectors on (Ω,Α,P)</li><li>Countably Infinite Product Probability Spaces</li><li>Random Elements and Processes on&nbsp;(Ω,Α,P)</li></ol></div>Chapter 6. Distribution and Quantile Functions<div><ol><li>Character of Distribution Functions</li><li>Properties of Distribution Functions</li><li>The Quantile Transformation</li><li>Integration by Parts Applied to Moments</li><li>Important Statistical Quantities</li><li>Infinite Variances</li></ol></div><div>Chapter 7. Independence and Conditional Distributions</div><div><ol><li>Independence</li><li>The Tail&nbsp;σ-Field</li><li>Uncorrelated Random Variables</li><li>Basic Properties of Conditional Expectation</li><li>Regular Conditional Probability</li></ol></div><div>Chapter 8. WLLN, SLLN, LIL, and Series</div><div><ol><li>Introduction</li><li>Borel Cantelli and Kronecker Lemmas</li><li>Truncation, WLLN, and Review of Inequalities</li><li>Maximal Inequalities and Symmetrization</li><li>The Classical Laws of Large Numbers (or, LLNs)</li><li>Applications of the Laws of Large Numbers</li><li>Law of the Iterated Logarithm (or, LIL)</li><li>Strong Markov Property for Sums of IID RVs</li><li>Convergence of Series of Independent RVs</li><li>Martinagles</li><li>Maximal Inequalities, Some with&nbsp;↗ Boundaries</li></ol></div>Chapter 9. Characteristic Functions and Determining Classes<div><ol><li>Classical Convergence in Distribution</li><li>Determining Classes of Functions</li><li>Characteristic Functions, with Basic Results</li><li>Uniqueness and Inversion</li><li>The Continuity Theorem</li><li>Elementary Complex and Fourier Analysis</li><li>Esseen's Lemma</li><li>Distributions on Grids</li><li>Conditions for&nbsp;Ø to Be a Characteristic Function</li></ol></div><div>Chapter 10. CLTs via Characteristic Functions</div><div><ol><li>Introduction</li><li>Basic Limit Theorems</li><li>Variations on the Classical CLT</li><li>Examples of Limiting Distributions</li><li>Local Limit Theorems</li><li>Normality Via Winsorization and Truncation</li><li>Identically Distributed RVs</li><li>A Converse of the Classical CLT</li><li>Bootstrapping</li><li>Bootstrapping with Slowly&nbsp;↗ Winsorization</li></ol></div>Chapter 11. Infinitely Divisible and Stable Distributions<div><ol><li>Infinitely Divisible Distributions</li><li>Stable Distributions</li><li>Characterizing Stable Laws</li><li>The Domain of Attraction of a Stable Law</li><li>Gamma Approximations</li><li>Edgeworth Expansions</li></ol></div><div>Chapter 12. Brownian Motion and Empirical Processes</div><div><ol><li>Special Spaces&nbsp;</li><li>Existence of Processes on (C, C) and (D, D)</li><li>Brownian Motion and Brownian Bridge</li><li>Stopping Times</li><li>Strong Markov Property</li><li>Embedding a RV in Brownian Motion</li><li>Barrier Crossing Probabilities</li><li>Embedding the Partial Sum Process</li><li>Other Properties of Brownian Motion</li><li>Various Empirical Processes</li><li>Inequalities for the Various Empirical Processes</li><li>Applications</li></ol></div><div>Chapter 13. Martingales</div><div><ol><li>Basic Technicalities for Martingales</li><li>Simple Optional Sampling Theorem</li><li>The Submartingale Convergence Theorem</li><li>Applications of the S-mg Convergence Theorem</li><li>Decomposition of a Submartingale Sequence</li><li>Optional Sampling</li><li>Applications of Optional Sampling</li><li>Introduction to Counting Process Martingales</li><li>CLTs for Dependent RVs</li></ol></div><div>Chapter 14. Convergence in Law on Metric Spaces</div><div><ol><li>Convergence in Distribution on Metric Spaces</li><li>Metrics for Convergence in Distribution&nbsp;</li></ol></div><div>Chapter 15. Asymptotics Via Empirical Processes</div><div><ol><li>Introduction</li><li>Trimmed and Winsorized Means</li><li>Linear Rank Statistics and Finite Sampling</li><li>L-Statistics</li></ol></div><div>Appendix A. Special Distributions</div><div><ol>Elementary Probability<li>Distribution Theory for Statistics</li><div>Appendix B. General Topology and Hilbert Space</div><div><ol><li>General Topology</li><li>Metric Spaces</li><li>Hilbert Space</li></ol></div>Appendix C. More WLLN and CLT<div><ol><li>Introduction</li><li>General Moment Estimation Specific</li><li>Slowly Varying Partial Variance&nbsp;when&nbsp;σ2=∞</li><li>Specific Tail Relationships</li><li>Regularly Varying Functions</li><li>Some Winsorized Variance Comparison</li><li>Inequalities for Winsorized Quantile Functions</li></ol></div><div>References</div><div>Index</div></ol></div>