MSF200/MVE330 Stochastic Processes

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

Integration Theory

Weak Convergence