- This is a webpage for a 7.5 hp course
starting on Tuersday 18 March 2014 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.**

- 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.

- Mondays 15.15-17.00, room
**MVH12**(except May 12) - Wednesdays 10.00-11.45, room
**MVH12**(except April 9)

- Lecture 1. Borel-Cantelli lemmas. Modes of convergence of random variables.
- Lecture 2. Inequalities involving expectations. 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.

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.

- 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 date: 3 June 2014, 08:30-12:30, Vagg och Vatten
- Reexam date: 27 August 2014, 14:00-18:00, Johanneberg
- 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.

- Exam-2014, Exam-2013

- List-2014, List-2013

- Integration Theory
- Weak Convergence