Mathematical foundations of probability

(MASM14)

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 2017, 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:                    October 31,   2017,

                                                       time:     10:15 -12:00

                                                     place:      MH 227
 

Further schedule:    

Wednesday 1/11 8-10 in MH:227

Thursday 2/11 in MH:227

NO LECTURES ON NOVEMBER 7-8

Starting November 9, lectures run 3 times a week, 2 hours each :

Tuesday 10-12 in MH:227

Wednesday 8-10 in MH:227

Thursday 10-12 in MH:227



The last lecture is on December 14


CURRENT INFORMATION

EXAM(ORAL):

January ... MH:228

Home work: Chapter II: read pages: 131-176

Problems: 1: 1-4, 2: 1-3, 3: 1, 8, 9, 4: 1, 2,4,5,6,7


 
 

             

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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 $-algebras, the measurable spaces (${\mathbb R}$,${\cal B}({\mathbb R}$)),

$({\mathbb R}^n,{\cal B}({\mathbb R}^n) )$ and $({\mathbb R}^{\infty},{\cal B}({\mathbb R}^{\infty}) )$.

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 (${\mathbb R^{\infty}},{\cal B}({\mathbb R^{\infty}}$) )

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.