Course Information

Course Title	Code	Semester	L+U Hour	Credits	ECTS
ADVANCED PROBABILITY THEORY	801000805170		3 + 0	3.0	10.0

Prerequisites

None

Language of Instruction	Turkish
Course Level	Graduate Degree
Course Type	Compulsory
Mode of delivery	Oral Presentation
Course Coordinator
Instructors	Halil AYDOĞDU
Assistants
Goals	Adapt to fundamental concepts of probability theory such as probability spaces, limit theorems
Course Content	Probability spaces, Lebesgue integral, Radon-Nikodym theorem, characteristic function, convergence, limit theorems, stable distributions, infinity divisible distributions, ergodic theorems, martingales.
Learning Outcomes	1) Recognizes probability space and the Lebesgue integral 2) Understand the Radon-Nikodym theorem 3) Associate notion of convergence with the limit theorems 4) Recognizes stable distributions, infinite divided distributions 5) Understands the Martingale's.

Week	Topics	Teaching and Learning Methods and Techniques	Study Materials
1. Week	Measure Spaces, Lebesgue measure, measurable functions	Lecture	Homework
2. Week	Lebesgue integral, Fubini Theorem	Lecture	Homework
3. Week	Derivative of the integrals, Radon-Nikodym Theorem	Lecture	Homework
4. Week	Probability spaces, random variables, random vectors and random elements	Lecture; Question Answer; Discussion	Homework
5. Week	Discrete, continuous, singular and mixed distributions	Lecture	Homework
6. Week	Conditional Distributions, Independence of Random Variables	Lecture; Question Answer	Homework
7. Week	Expected Value, Characteristic Functions	Lecture; Question Answer	Homework
8. Week	Markov, Chebyshev, Jensen, Holder, Schwarz, Lyapunov inequalities. Convergence in Sequences of Random Variables	Lecture	Homework
9. Week	Law of Large Numbers, Central Limit Theorem, Multivariate Central Limit Theorem	Lecture; Discussion	Homework
10. Week	Delta Method, Asymptotic Distributions	Lecture	Homework
11. Week	Stable distributions, infinite divided distributions	Lecture; Case Study	Homework
12. Week	Conditional Expectation, Martingale's	Lecture	Homework
13. Week	Stochastic Processes, Finite-Dimensional Distributions, Kolmogorov ExistenceTheorem	Lecture	Homework
14. Week	Brownian Motion, Wiener Process	Lecture	Homework

Recommended Sources

Ash, R.B., Doléans-Dade, C. A. (1999). Probability & Measure Theory, Academic Press; 2 edition

Billingsley, P. (1986). Probability and Measure,John Wiley and Sons.

Royden, H.L. (1968). Real Analysis, MACMILLAN Publ.Co.,INC.

*DK = Course's Contrubution.

	0	1	2	3	4	5
Level of contribution	None	Very Low	Low	Fair	High	Very High

Event	Quantity	Duration (Hour)	Total Workload (Hour)
Course Duration (Total weeks*Hours per week)	14	8	112
Work Hour outside Classroom (Preparation, strengthening)	14	8	112
Homework	3	7	21
Midterm Exam	1	6	6
Time to prepare for Midterm Exam	1	20	20
Final Exam	1	6	6
Time to prepare for Final Exam	1	20	20
Total Workload
Total Workload / 30 (s)
ECTS Credit of the Course			30