Download Schedule for Probability Theory IV, 7

Schedule for the course
Probability Theory IV, 7.5 ECTS
credits, Spring 2014
Day 1, Tuesday, January 21, Room 31, House 5, 13.15-16.00
Introduction to the course (one hour).
Lecture 1 (two hours): Measurable Spaces (algebras, -algebras, monotone classes,
Borel -algebras, Rk, metric spaces, functional spaces).
Literature: R 1 + 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 28, 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 11, Room 31, House 5, 13.15-16.00
Lecture 3 (two hours): Probability Measures - 2 (continuation theorem, distribution
functions, decomposition of distribution functions, -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 18, 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 25, 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 4, 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 11, 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 18, 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, March 25, 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 1, 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, April 8, 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, April 15, 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 6, 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, Tuesday, May 13, Room 31, House 5, 13.15-16.00
Seminar (presentation of student’s reports) (3 hours).
Day 15, Tuesday, May 20, Room 31, House 5, 13.15-16.00
Written Test (3 hours).
------------------------------------------------------------------------------------------------------
Course materials
R: Resnik, S.I. A Probability Path, Birkhäuser, 1998.
LN: Lecture Notes for the course (copies of transparencies used at the lectures).
G: Gut, A. Probability: A Graduate Course, Springer, 2005 [supplementary].