Last updated 22 March 2016 MSF200/MVE330 Stochastic Processes

MSF200/MVE330 Stochastic Processes

This is a webpage for a 7.5 hp course starting on Tuersday 22 March 2016 at 10.00 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)
Tuesdays 10.00-11.45, room MVH12
Fridays 15.15-17.00, room MVH12 (except 25.03, 6.05, 27.05)
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: 3 June 2016, 08:30-12:30. Please, register before 12 May 2016.
Cheat sheet: on the final exam the student may use two 2-sided A4-pages with a course digest.
This digest should be put together by each of the student themself and brought with them to the final exam.
Old exams
Exam-2014, Exam-2013
Lists of students
List-2016, List-2014, List-2013
Related courses
Integration Theory
Weak Convergence