The Mathematics of Chance / ContentsPage progress
Proof Scope of the Early Six Chapters
This table records the historical proof scope of the early six-chapter manuscript, not the current book’s completion state. Chapters 11–12 and the foundation companions provide later proofs. Consult the topic index for current ownership and pending results.
G. The Limits of the Proofs of Deeper Theorems
| Location in the text | What this book actually supplies | What is still invoked as a known result |
|---|---|---|
| Chapter 2: long-run frequencies and posteriors | Finite-variance weak law of large numbers; specific beta–Bernoulli calculations | The reference to the strong law does not include a complete proof of the general strong law |
| Chapter 3: characteristic-function approach | Second-order expansion and product limit; Appendix D supplies remainder control | Lévy's continuity theorem and uniqueness of characteristic functions; the integral calculation of the Cauchy characteristic function remains a known example |
| Section 4.7 | Interpolation, path space, and the distinction between finite-dimensional convergence and tightness | The complete tightness proof for Donsker's invariance principle |
| Section 4.8 | Consistent finite-dimensional distributions and a fourth-moment continuity estimate | Complete proofs of the Kolmogorov extension and continuous-modification theorems |
| Section 4.10 | Increment scaling, quadratic variation, and its conflict with smoothness | The complete proof of almost-sure nowhere differentiability; an argument at a fixed time cannot be applied directly across uncountably many times |
| Section 4.11 | The pathwise idea of reflection and the maximum formula | Measure preservation of the reflection transformation can be established using the strong Markov property; the strong Markov theorem itself is not proved in the six chapters |
| Section 5.8 | The original proof strategy; Appendices A and B supply closure, additivity, and treatment of cover endpoints | The listed gaps have now been filled here |
| Section 6.12 | The original route through rectangles, simple functions, and convergence; Appendix C supplies the intermediate set argument | The \pi-\lambda theorem is invoked as a set-theoretic tool |