**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.

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**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: **

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EXAM(ORAL):

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

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Content of the course MASM14
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1. $\sigma $-algebras, measures, measurable space.
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Theorem on the equivalent conditions for the finitely additive set function
to be a
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probability.
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2. Borel $\sigma $-algebras, the measurable spaces (${\mathbb R}$,${\cal B}({\mathbb R}$)),
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$({\mathbb R}^n,{\cal B}({\mathbb R}^n) )$ and $({\mathbb R}^{\infty},{\cal B}({\mathbb R}^{\infty}) )$.
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3. Probability measures on measurable spaces. Theorem on one-to-one
correspondence
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between the probabilities and the distributions.
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4. Measures: discrete, continuous, singular. An example of a singular measure.
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Kolmogorov's Theorem on the extension of measures in
(${\mathbb R^{\infty}},{\cal B}({\mathbb R^{\infty}}$) )
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5. Random variables. Lemma: Borel function of a random variable is a random
variable.
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6. Theorem on the limits of the sequences of the extended random variables.
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7. $F_{\xi}$ $\sigma $-algebra. Theorem on the representation of the
$F_{\xi}$-measurable
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random variable.
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8. Random elements. Definition of the independence. The necessary and sufficient
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conditions of the independence.
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9. Lebesgue integral: definition, properties.
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10. Theorem on monotone convergence.
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11. Fatou's Lemma.
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12. Lebesgue's Theorem on Dominated convergence.
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13. Chebyshev's, the Cauchy-Bunyakovskii and Jensen's inequalities.
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14. Lyapunov's, Hölder's and Minkowski's inequalities.
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15. Theorem on change of variables in a Lebesgue integral.
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16. Fubini's theorem.
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17. Conditional expectation with respect to a $\sigma $-algebra: definition
and properties.
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18. Theorem on taking limits under the expectation sign.
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19. Characteristic functions. Uniqueness theorem.
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20. Helly's Theorem.
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21. Prokhorov's Theorem.
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22. Continuity Theorem.
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