Probability IV

Day 1, Tuesday, January 19, Room 31, House 5, 13.15-16.00. Introduction to the course (one hour). Lecture 1 (two hours): Measurable Spaces (algebras, s -algebras, monotone classes, Borel s -algebras, R_k, metric spaces, functional spaces). Literature: R +  G 1.1, 1.2. Recommended problems: R 1.9: 3 – 12, 15 – 23, 27, 29, 30, 32, 34, 41, 43, 44; LN problems + G 1.6: 1 – 4, 6 – 8.

Day 2, Tuesday, January 26, Room 31, House 5, 13.15-16.00. Lecture 2 (two hours): Probability Measures - 1 (definition, basic properties, probability spaces, independence, conditional probabilities). Problem solving (one hour). Literature: R 2.1, 2.3 + G 1.3, 1.4. Recommended problems: R 2.6: 1 – 6; LN problems + G 1.6: 5, 9 – 12.

Day 3, Tuesday, February 2, Room 31, House 5, 13.15-16.00. Lecture 3 (two hours): Probability Measures - 2 (continuation theorem, distribution functions, decomposition of distribution functions, s -finite measures, Lebesgue measure). Problem solving (one hour). Literature: R 2.2, 2.4, 2.5 + G 2.2. Recommended problems: R 2.6: 9 – 18, 20 – 23; LN problems + G 1.6: 5, 13.

Day 4, Tuesday, February 16, Room 31, House 5, 13.15-16.00. Lecture 4 (two hours): Random Variables (random variables, random vectors, random elements, transformations of random variables, distribution functions, measures generated by random variables). Problem solving (one hour). Literature: R 3.1 – 3.3 + G 2.1, 2.3, 2.10, 2.13 – 2.15. Recommended problems: R 3.4: 1 – 9, 11 – 20, 25; LN problems + G 2.20: 1 – 4, 7, 9, 10.

Day 5, Tuesday, February 23, Room 31, House 5, 13.15-16.00. Lecture 5 (two hours): Expectations (definition, basic properties, high order moments, inequalities). Problem solving (one hour). Literature: R 5.1, 5.2 + G 2.4, 2.8, 2.12, 3.1 – 3.5. Recommended problems: R 5.10: 1 – 5 + 22, 24, 29, 31, 33, 40; LN problems + G 2.20: 13, 14, 16 – 19, 22.

Day 6, Tuesday, March 2, Room 31, House 5, 13.15-16.00. Lecture 6 (two hours): Expectations and Lebesgue integration (exchange of variables, integration with absolute continuous measures, Lebesgue and Riemann integration, Lebesgue theorem, Fubini theorem). Problem solving (one hour). Literature: R 5.3 – 5.9 + G 2.5, 2.7 – 2.9. Recommended problems: R 5.10: 6 – 11, 13, 16 – 20, 37; LN problems + G 2.20: 13, 14, 16 – 19, 22.

Day 7, Tuesday, March 16, Room 31, House 5, 13.15-16.00. Lecture 7 (2 hours): Strong Limit Theorems (Borel-Cantelli lemmas, 0-1 Kolmogorov law, a.s. convergence and convergence in probability, convergence of random series, strong and weak laws of large numbers, applications). Problem solving (one hour). Literature: R 4.5, 6.1 – 6.3, 7 + G 2.10, 2.18, 6.1 – 6.6. Recommended problems: R 6.8: 1, 2, 11 – 17, 24, 25, 31, 33, R 7.7: 1, 2, 4, 14, 19, 21; LN problems + G 5.14: 1, 10, 11, G 6.13: 1, 2, 7, 11, 12, 17 + 21.

Day 8, Tuesday, March 23, Room 31, House 5, 13.15-16.00. Lecture 8 (2 hours): Weak Convergence -1 (definitions, connection with other type of convergence, Skorokhod representation theorem, weak convergence and expectations). Problem solving (one hour). Literature: R 8.1, 8.3 – 8.5 + G 5.1 – 5.5. Recommended problems: R 8.8: 1 – 3, 7, 10, 31, 32, 37; LN problems + G 5.14: 3 – 7, 12, 20, 21, 23.

Day 9, Tuesday, April 6, Room 31, House 5, 13.15-16.00. Lecture 9 (2 hours): Weak Convergence - 2 (subsequence approach to weak convergence, weak convergence of transformed random variables, functional limit theorems). Problem solving (one hour). Literature: R 9.6 + G 5.7, 5.8, 5.10, 5.13. Recommended problems: R 8.8: 4, 5, 11, 19, R 9.9: 2, 11, 20, 21, 32; LN problems + G 5.14: 24, 25.

Day 10, Tuesday, April 13, Room 31, House 5, 13.15-16.00. Lecture 10 (2 hours): Characteristic Functions (definition, Bochnar theorem, Parseval identity, inversion formulas, regularity properties, continuity theorem, central limit theorem, limit theorems for random sums and other applications). Problem solving (one hour). Literature: R 9.1 – 9.5, 9.7, 9.8 + G 4.1 – 4.5, 4.9. Recommended problems: R 9.9: 1 – 3, 5, 6, 9, 10, 14, 15, 22, 28, 33, 35; LN problems + G 4.10: 1, 8, 10, 21, 22, 24.

Day 11, Tuesday, May 4, Room 31, House 5, 13.15-16.00. Lecture 11 (2 hours): Limit Theorems (infinitely divisible distributions, general limit theorems for sums of independent random variables, extreme distributions, limit theorems for extremes). Problem solving (one hour). Literature: R 8.7 + G 9.1 – 9.4, 9.6. Recommended problems: R 8.8: 1, 2, 3, 7; 16, 17, 27, 31, 34; LN problems + G 9.8: 7 – 13.

Day 12, Tuesday, May 11, Room 31, House 5, 13.15-16.00. Lecture 12 (2 hour): Conditional Expectations (definition, Radon-Nikodym theorem, basic properties, applications). Problem solving (one hour). Literature: G 10.1; R 10.1 – 10.3. Recommended problems: R 10.17: 1, 2, 7, 11, 12; LN problems + G 2.6: 2, 9, 10, 11, 12, 13, 24, 25, 45 + G 10.17: 1, 2, 5, 7 – 9.

Day 13, Tuesday, May 18 Room 31, House 5, 13.15-16.00. Lecture 13 (2 hours): Martingales (definition, basic properties, stopping times, convergence, applications). Literature: G 10.2 – 10.10; R 10.4 – 10.7, 10.14, 10.15. Recommended problems: R 10.17: 14 – 17, 22, 28, 41, 51, 52, 55; LN problems + G 10.17: 11 – 13, 20, 25.

Day 14, Thursday, May 20, Room 31, House 5, 13.15-16.00. Survey of Topics for Further Studies (1 hour). Oral Examination ( 2 hours).

Day 15, Thursday, June 3, Room 31, House 5, 13.15-16.00. Written Test (3 hours).