| Lecture | Date | Topics Discussed | See |
|---|---|---|---|
| 1 | Wed, Aug 25. | Administrative details. Definition of measure space, measurable functions, etc, and corresponding probability terms: probability space, random variable. | Durrett 1.1 and 1.3, or any measure theory textbook. |
| 2 | Fri, Aug 27. | Random vectors, operations on random variables (arithmetic, applying measurable functions, inf/sup/limits, almost sure convergence). Distribution of a random variable (a probability measure on R); examples of common discrete and continuous distributions. Joint distribution of several random variables. | Durrett 1.1, 1.2, 1.3 |
| 3 | Mon, Aug 30. | Distribution functions for random variables. Definition of the Lebesgue integral on an abstract measure space, basic properties, convergence theorems. Hölder's inequality. | Durrett 1.4, 1.5, 1.6. |
| 4 | Wed, Sep 1. | Lp norms and convergence. L1 convergence does not imply almost sure convergence nor vice versa. Convex functions and Jensen's inequality. Chebyshev's inequality. | Durrett 1.5, 1.6 |
| 5 | Fri, Sep 3. | Expectation in terms of distribution. Some philosophy about σ-fields and measurability, interpreted in terms of "information." The σ-field generated by a collection of sets. | Durrett 1.6.3; S. Resnick, A Probability Path (on reserve at Mathematics Library) has more details on σ-fields and measurability in Chapters 1 and 3. |
| - | Mon, Sep 6. | Labor Day, no class. | |
| 6 | Wed, Sep 8. | Dynkin's π-λ theorem. Application: a distribution μ is uniquely determined by its distribution function F. The σ-field generated by one or more random variables. | Durrett A.1.4; Durrett 1.3; Resnick Chapter 3. |
| 7 | Fri, Sep 10. | Independence: for events, σ-fields, random variables. Why pairwise independence is not strong enough. Deterministic events are independent of everything. Product measures and the Tonelli/Fubini theorems. | Durrett 2.1, 1.7 |
| 8 | Mon, Sep 13. | Fire alarm. Independent random variables correspond to product measures; expectations of products of independent random variables factor. | Durrett 2.1.2 |
| 9 | Wed, Sep 15. | Independent π-systems generate independent σ-fields. Disjoint subsets of a set of independent random variables generate independent σ-fields. Kolmogorov's extension theorem, and the special case that sequences of independent random variables with any given distributions exist. | Durrett 2.1.1, 2.1.4. Kolmogorov extension theorem is A.3.1. |
| 10 | Fri, Sep 17. | Construction of a sequence of independent random variables with any given distributions. Start with a uniform distribution; extract its bits to get an iid sequence of coin flips; regroup to get a sequence of iid sequences, all independent; reassemble to get an iid sequence of uniforms; transform to get desired distributions. | Inspired by Durrett exercise 2.1.18. |
| 11 | Mon, Sep 20. | Borel-Cantelli lemmas, Borel 0-1 Law, statement of Kolmogorov 0-1 Law. | Durrett 2.3, 2.5. |
| 12 | Wed, Sep 22. | Proof of Kolmogorov 0-1 law. Definition and some basic notions about convergence in probability. | Durrett 2.5, 2.2, 2.3. |
| 13 | Fri, Sep 24. | Some more facts about convergence in probability. The sub-subsequence trick. An L2 weak law using Chebyshev and an L4 strong law of large numbers. | Durrett 2.2, 2.3. |
| 14 | Mon, Sep 27. | Some motivation for the difference between weak and strong laws of large numbers. Overview of various versions of LLN. Applications: Glivenko-Cantelli theorem, normal numbers. | Glivenko-Cantelli: Durrett Theorem 2.4.7. Normal numbers: see section 7 of J.A. Goldstein, "Some applications of the law of large numbers," Bol. Soc. Brasil. Mat. 6 (1975), no. 1, 25-38. |
| 15 | Wed, Sep 29. | Intro to stochastic processes. Definition of filtration, adapted and predictable process. Gambling systems as an example. The original martingale, where you double your bet when you lose. Conditional expectation: proof of uniqueness and linearity (existence to come). | Definitions: Durrett pages 232, 234. Conditional expectation: Durrett 5.1. |
| 16 | Fri, Oct 1. | Further properties of conditional expectation. Skipped the proof of existence. | Durrett 5.1.2. For some notes on the existence of conditional expectation, see here. |
| 17 | Mon, Oct 4. | Guest lecture (Jon Peterson) | |
| 18 | Wed, Oct 6. | Guest lecture (Jon Peterson) | |
| 19 | Fri, Oct 8. | Guest lecture (Jon Peterson) | |
| - | Mon, Oct 11. | Fall break, no class. | |
| 20 | Wed, Oct 13. | Upcrossing inequality, martingale convergence theorem. | Durrett Theorems 5.2.7, 5.2.8, 5.2.9. |
| 21 | Fri, Oct 15. | Martingales with bounded increments converge or oscillate widely. Smartingales that converge a.s. need not converge in L1. Branching process: subcritical extinction. | Durrett Theorem 5.3.1, Example 5.2.3, Exercise 5.2.4, Lemma 5.3.6, Theorem 5.3.7. |
| 22 | Mon, Oct 18. | Branching process: critical extinction. Definition of uniform integrability, statement of Vitali convergence theorem. A single integrable random variable is ui, as is a family dominated by another ui family. Alternate criteria for uniform integrability (uniform absolute continuity + L1 boundedness). | Branching process: Durrett Theorem 5.3.8. One proof of supercritical nonextinction is Theorem 5.3.9; we will give a different proof next class or so. Uniform integrability: section 5.5. Vitali convergence theorem: Theorem 5.5.2. Alternate criteria: see Resnick, A Probability Path, Theorem 6.5.1 (on reserve at Mathematics Library), or section 12.7 of Bruce Driver's notes. |
| 23 | Wed, Oct 20. | A criterion for uniform integrability (Durrett Exercise 5.5.1). Proof of Vitali convergence theorem. Consequences for uniformly integrable smartingales. | Vitali convergence theorem: Theorem 5.5.2. Smartingales: Theorem 5.5.3. |
| 24 | Fri, Oct 22. | Branching process: supercritical survival. Conditioning gives uniform integrability. A uniformly integrable martingale has a "last element". | Branching process: like Durrett Example 5.4.3, except we used Exercise 5.5.1 and Theorem 5.5.3 (giving convergence in L1) instead of Theorem 5.4.5 (convergence in L2). Conditioning: Theorem 5.5.1. Ui martingales: Lemma 5.5.5, Theorem 5.5.6, Theorem 5.5.7. |
| 25 | Mon, Oct 25. | Backward martingales. Beginning of backward martingale proof of strong law of large numbers. | Durrett Theorems 5.6.1, 5.6.2, Example 5.6.1. |
| 26 | Wed, Oct 27. | End of backward martingale proof of SLLN. Baby optional stopping theorems. | Durrett Example 5.6.1, Theorems 5.4.1, 5.7.1. |
| 27 | Fri, Oct 29. | Optional stopping theorems. Application to asymmetric simple random walk. | Durrett Theorems 5.7.1 thru 5.7.4 (first part), 5.7.7. |
| 28 | Mon, Nov 1. | More computations for asymmetric simple random walk. Additional criteria for optional stopping. | Durrett Theorems 5.7.7, 5.7.5. |
| 29 | Wed, Nov 3. | A trivial optional stopping inequality for positive supermartingales. Stopped σ-fields; submartingale property preserved at stopping times under ui assumption. | Durrett Theorem 5.7.6, Exercises 4.1.5-4.1.7 (stopped σ-fields), Theorem 5.7.4 (second half). |
| 30 | Fri, Nov 5. | Doob decomposition theorem. Quadratic variation <X> of an L2 martingale X. Computation of Doob decompositions for discrete stochastic integrals. Relationship between finiteness of <X>∞ and convergence of Xn. | Doob decomposition: Durrett Theorem 5.2.10. L2 martingales: See D. Williams, Probability with Martingales, 12.12-12.14. |
| 31 | Mon, Nov 8. | Proof of theorem that Xn converges a.s. on {<X>∞ < ∞}. A SLLN for L2 martingales. Preview of central limit theorem. | D. Williams, Probability with Martingales, 12.12-12.14. |
| 32 | Wed, Nov 10. | Definition of weak convergence. Reason for the "continuity point" condition in the definition. Examples and counterexamples. Convergence i.p. implies weak convergence, and weak convergence to a constant implies convergence i.p. | Durrett section 3.2.1, exercise 3.2.12. |
| 33 | Fri, Nov 12. | Skorohod representation theorem (Durrett Theorem 3.2.2). Weak convergence is equivalent to convergence of expectations of bounded continuous functions, composition with continuous functions preserves weak convergence. | Durrett Theorems 3.2.2-3.2.4. |
| 34 | Mon, Nov 15. | Equivalent definitions of weak convergence ("Portmanteau theorem", Durrett Theorem 3.2.5). Vague convergence, tightness. A vaguely converging sequence that is tight converges weakly. | Durrett Theorems 3.2.5, 3.2.7 |
| 35 | Wed, Nov 17. | Helly selection theorem: Every sequence of distributions has a vaguely converging subsequence. Prohorov theorem: Tight sets of distributions are weakly compact (have weakly converging subsequences). Characteristic functions and basic properties. | Durrett Theorem 3.2.6, Theorem 3.3.1. |
| 36 | Fri, Nov 19. | Differentiation under expectation. Moments of a random variable correspond to derivatives of its chf. Computation of some chfs: Bernoulli, uniform, normal. Preliminary steps for uniqueness theorem (a distribution is uniquely determined by its chf): you can check weak convergence using only compactly supported functions, and Parseval's identity. | Durrett Theorem A.5.2 (similar), Exercise 3.3.14, Examples 3.3.1, 3.3.4, 3.3.3. Durrett's version of uniqueness theorem is Theorem 3.3.4; my proof is from Resnick, A Probability Path, Theorem 9.5.1. |
| 37 | Mon, Nov 22. | Proof of the uniqueness theorem. Fourier inversion; integrable chfs correspond to continuous densities. Start of the proof of the continuity theorem. | Uniqueness: Resnick, A Probability Path, Theorem 9.5.1. Durrett's version is Theorems 3.3.4, 3.3.5. Continuity: my proof comes from Pathak, P. K. Alternative proofs of some theorems on characteristic functions. Sankhya Ser. A 28 1966 309-314. Durrett's version is Theorem 3.3.6. |
| 38 | Wed, Nov 24. | Finish proof of continuity theorem. Prove classical central limit theorem (for sums of iid random variables with finite variance). | Durrett Theorem 3.4.1. Note that the approximation from Theorem 3.3.8 is just a simple consequence of Taylor's theorem. |
| - | Fri, Nov 26. | Thanksgiving break, no class. | |
| 39 | Mon, Nov 29. | Two derivatives on a chf implies finite second on the corresponding distribution, but one derivative does not imply first moment. CLT applications: normal approximation to binomial, Weldon's dice experiment and simple hypothesis testing. | Durrett Theorem 3.3.9, Example 3.4.3. |
| 40 | Wed, Dec 1. | Some complements: finite second moment is necessary for weak convergence with n1/2 scaling. The CLT cannot be strengthened to give convergence in probability. Statement and proof sketch for the Lindeberg-Feller CLT. | Durrett Exercises 3.4.2, 3.4.3, Theorem 3.4.5. |
| 41 | Fri, Dec 3. | The CLT in several dimensions. Covariance matrices, multivariate normal, weak convergence and chfs in several dimensions. | Durrett Section 3.9. |