Department:
Mathematical Statistics, Centre for Mathematical Sciences, Lund University
with Lund Institute of Technology
Credits:
Mon 5 (7.5 ECTS credits)
Lecturer:
Tatyana Turova, phone: +46 46 222 8543, office: 220-219, email: tatyana@maths.lth.se
Time period:
Fall 2011, period 2.
Course book:
The course will closely follow the book by A. N. Shiryaev, Probability, Springer 1996.
Prerequisites:
30 points (45 ECTS credits) in Mathematics. An introductory course in probability and a second course in probability are desirable
but not necessary.
FIRST MEETING: October 31, 2011,
time: 10:15 -12:00
place: MH 332 B
Further schedule: UPDATED!!!
CURRENT INFORMATION
Overview lecture and consultations: January 11, 10-12 and January 12, 10-12 in MH:227
Read in the book "Probability" by A.N.Shiryaev (1996): pp. 131-234,
Problems: Chapter II, §1, problems 1-4, 5-8, $2 problems 1, 2, 3, $3 problems 1, 4, 7, 8, 9. §4 problems 1-8, §6 problem 1, 2, 5, 6, 7, 8, 10, 11, 13, §7 problems 1, 2.
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Content of the course MASM14
1. $\sigma $-algebras, measures, measurable space.
Theorem on the equivalent conditions for the finitely additive set function to be a
probability.
2. Borel $\sigma $-algebras, the measurable spaces (R , B(R) ),
$({\bf R}^n , {\cal B}({\bf R}^n) )$ and $({\bf R}^{\infty}, {\cal B}({\bf R}^{\infty}) )$.
3. Probability measures on measurable spaces. Theorem on one-to-one correspondence
between the probabilities and the distributions.
4. Measures: discrete, continuous, singular. An example of a singular measure.
Kolmogorov's Theorem on the extension of measures in (R^{\infty} , B(R^{\infty}) )
5. Random variables. Lemma: Borel function of a random variable is a random variable.
6. Theorem on the limits of the sequences of the extended random variables.
7. $F_{\xi}$ $\sigma $-algebra. Theorem on the representation of the $F_{\xi}$-measurable
random variable.
8. Random elements. Definition of the independence. The necessary and sufficient
conditions of the independence.
9. Lebesgue integral: definition, properties.
10. Theorem on monotone convergence.
11. Fatou's Lemma.
12. Lebesgue's Theorem on Dominated convergence.
13. Chebyshev's, the Cauchy-Bunyakovskii and Jensen's inequalities.
14. Lyapunov's, Hölder's and Minkowski's inequalities.
15. Theorem on change of variables in a Lebesgue integral.
16. Fubini's theorem.
17. Conditional expectation with respect to a $\sigma $-algebra: definition and properties.
18. Theorem on taking limits under the expectation sign.
19. Characteristic functions. Uniqueness theorem.
20. Helly's Theorem.
21. Prokhorov's Theorem.
22. Continuity Theorem.