# 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:

### NO LECTURES ON NOVEMBER 7-8

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

### CURRENT INFORMATION

EXTRA SEMINARS: December 18, time 10-12, place MH: 362A December 19, time 10-12, place MH: 362A December 20, time 13-15, place MH: 362A

EXAM(ORAL) !!! NEW TIME !!!!:

### January 24 : 12:30 -15 in MH:228

Home work: Chapter II: read pages: 131-230, 274-286 (§12), Chapter III: read pages: 308-325.

### Problems: §1: 1-4, §2: 1-3, §3: 1, 8, 9, §4: 1, 2,4,5,6,7, §5: 1, §6: 2, 4, 6, 8, 11, 13, §7: 1,2 §12: 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.