| Lecture # |
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Topic-Main |
Topic-Sub |
Available Sources* |
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| 1 |
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Introduction and Motivation |
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W.0 |
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| 2 |
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Probability (Measure) spaces |
Definition, Construction. |
W.1, D.A.1. |
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| 3 |
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Examples: Lebesgue and Stieltjes Measures. |
W.1, D.1.2. |
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| 4 |
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Proof of Caratheodory's thm. |
W.A1, D.A.1. |
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| 5 |
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Uniqueness, Completion, Set Operations, Borel-Cantelli
Lemma 1 |
W.A1, W.2, D.A.1-2, D.2.3. |
| 6 |
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Distribution Functions (on R), Absolute continuity,
Radon-Nykodym Derivative (no proof). Densities (on R). |
D.1, D.A.4. |
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| 7 |
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Random Variables (Measurable Functions) |
Measurability, Operations, Induced Measure (Distribution). |
W.3, D.1.2. |
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| 8 |
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Examples: Standard distributions, Cantor Function. |
? |
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| 9 |
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Expectation (Integration) |
Construction of the integral. |
W.5, D.1.4-5. |
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| 10 |
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Calculating expectation on R. |
D.1.6. |
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| 11 |
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Exchanging limiting and integration (Convergence Thms). |
W.5, D.1.6. |
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| 12 |
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Jensen, Markov, Schwartz, Holder, Minkowski, L^p Spaces. |
W.6, D.1.5-6. |
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| 13 |
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Types of Convergence and related theorems. |
? |
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| 14 |
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Independence |
Defintion (sigma algebras, random variables). |
W.4, D.2.1. |
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| 15 |
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Borel Cantelli Lemma 2. Kolomogorov's 0-1 law. |
W.4, D.2.3. |
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| 16 |
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Product Measures. Fubini's Theorem. |
W.8, D.1.7. |
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| 17 |
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Laws of Large numbers |
Weak Law of Large Numbers |
V.3, D.2.2. |
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| 18 |
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Kolmogorov's 3 series Theorem. |
V.3, D.2.5. |
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| 19 |
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Strong Law of Large Numbers. |
V.3, D.2.4-5. |
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| 20 |
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Weak Convergence |
Definition, Skorohod's Thm, Relation to other
convergences, Continuity. |
W. 17, V.2., D.3.2. |
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| 21 |
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Tightness. Metrizability of the weak topology. |
W.17, D.3.2. |
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| 22 |
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Examples. Other notions of convergence: Total Variation.
Wasserstein Metric. |
? |
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| 23 |
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Characteristic Functions |
Definition. Operations. Inversion. Other transforms. |
W. 16, V.2, D.3.3. |
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| 24 |
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Levy's convergence Theorem, Bochner's Theorem (no proof). |
W. 18, V.2, D.3.3. |
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| 25 |
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Central Limit Theorems |
Normal Convergence - IID case, ID case (Lindenberg
condition). |
D.3.4, W.18, V.3. |
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| 26 |
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Poission Convergence. |
D.3.6, V.3. |
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| 27 |
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Local CLT (no proof) and Cramer's theorem (with proof). |
D.3. |
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| 28 |
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Conditional Expectation |
Definition.
Existence. Examples. |
W.9, D.5.1., V.4. |
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| 29 |
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Properties. Independence. Regular Conditional
Probabilities. |
W.9, D.5.1., V.4. |
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| 30 |
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Stable Laws and Infinitely divisible
distributions. |
D.3.7-8. |
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* |
D - Durrett |
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W - Williams. |
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V - Varadhan. |
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