This is a webpage for a 7.5 hp course starting on
Wednesday 21 March 2018 at 15.15 in MVH12.

This
graduate course is aimed at the Master and PhD students at the Department of
Mathematical Sciences.

Main
topics: **Convergence, Stationarity, Renewals, Queues, Martingales.**

**The course textbook**

Main:
"Probability and Random Processes", 3rd edition, by Grimmet and Stirzaker. Chapters
7-12.

Continuously
updated lecture notes (**download**).
Chapters 2.2, 2.4, 3, 6-8.

Optional:
"One thousand exercises in probability" by Grimmet
and Stirzaker.

**Instructor:** Serik Sagitov **Time table
for 14 lectures (weeks 12, 15-22)**

Wednesdays
15.05-16.45, room **MVH12**

Fridays
15.05-16.45, room **MVH12** (except 11.05, 25.05)

**Course
content**

The simplest example of a stochastic process is the sequence of independent and
identically distributed random variables.

The classical results for this model are the Law of Large Numbers and the
Central Limit Theorem.

The fundamental models of stochastic processes considered in this course are
extensions of the classical IID setting.

Detailed list of topics

Lecture
1. Borel-Cantelli lemmas. Inequalities involving
expectations. Modes of convergence of random variables.

Lecture
2. Weakly and strongly stationary processes. Linear prediction.

Lecture
3. Spectral representation for weakly stationary processes.

Lecture
4. Ergodic theorems for stationary processes.

Lecture
5. Renewal function and excess life.

Lecture
6. Stopping times and Wald's equation.

Lecture
7. Regeneration techniques for queues.

Lecture
8. M/M/1 and M/G/1 queues.

Lecture
9. G/M/1 and G/G/1 queues.

Lecture
10. Martingales. Convergence in L^2.

Lecture
11. Doob's decomposition. Hoefding's
inequality.

Lecture
12. Convergence in L^1. Doob's martingale.

Lecture
13. Optional sampling theorem.

Lecture
14. Maximal inequality. Backward martingales. Course overview.

**Recommended exercises**
(some of them could be included to the final exam)

7.1.5,
7.2.1, 7.2.7, 7.3.1, 7.3.3, 7.3.9, 7.4.1.

7.5.1,
7.7.3, 7.8.3, 7.9.1, 7.10.6, 7.11.27.

9.1.2,
9.2.1, 9.2.2, 9.3.2, 9.3.3, 9.3.4, 9.4.2.

9.4.3,
9.5.2, 9.6.2, 9.6.4, 9.7.9, 9.7.12.

10.1.4,
10.2.1, 10.6.1, 11.3.1, 11.3.2, 11.6.1.

7.7.1,
7.7.2, 12.1.7, 12.1.8, 12.1.9, 12.2.1 (Doob's
martingale).

12.4.1,
12.4.5, 12.5.4, 12.9.6, 12.9.7, 12.9.13, 12.9.20.

**Final exam**

Final
exam date: 1 June 2018, 08:30-12:30.

Course
digest: on the final exam, it is expected that you use your own four A4-page
course digest.

Attach your digest report to the exam solutions – if
the report is generated by Latex and appropriately summarizes the course, you
may get a bonus point for it.

**Old exams**

Exam-2018, Exam-2016, Exam-2014, Exam-2013

**Lists of students**

List-2018, List-2016, List-2014, List-2013

**Related courses**