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 2012, period 2.
Course book:
The course will closely follow the book by A. N. Shiryaev, Probability, Springer 1996 (or any other edition).
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: November 6, 2012,
time: 10:15 -12:00
place: MH 228
Further schedule:
CURRENT INFORMATION
Read in the book "Probability" by A.N.Shiryaev (1996): pp. 1-215
Problems: Chapter II, §1 problems 1-5, §2 problems 1-2 §3 problems 1-9, §4 problems 1-6, §5 problem 1, §6 problems 1-13, §7, problem 7,
____________________________________________
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.