STATS 721 : Foundations of Stochastic Processes
2022 Semester Two (1225) (15 POINTS)
This course looks at the theory of stochastic processes, showing how complex systems can be built up from sequences of elementary random choices. In particular, it will present the theory and techniques of Markov chains which can be used as probability models in many diverse applications. The course may be useful for students with interests in Probability, Mathematics, Statistics, Operations Research, Finance, Engineering, Economics, and Theoretical Biology. Before taking this course students should have a good background in probability (at least Grade B in one of STATS125, STATS210, STATS225, or STATS320) as well as some mathematics (one of MATHS208, MATHS250, MATHS253, or equivalent). This course will also provide good preparation for more advanced courses in probability (such as STATS710, STATS720, or STATS723).
Capabilities Developed in this Course
|Disciplinary Knowledge and Practice
|Communication and Engagement
|Independence and Integrity
- Apply the techniques and constructions of discrete and continuous time Markov chains to solve problems involving n-step transition probabilities, hitting probabilities, and stationary distributions. (Capability 1, 2 and 3)
- Distinguish between transient and recurrent states in given finite and infinite Markov chains. (Capability 1 and 3)
- Translate a concrete stochastic process into the corresponding Markov chain given by its transition probabilities or rates. (Capability 1, 2 and 3)
- Apply generating functions to identify important features of Markov chains. (Capability 1 and 3)
- Organise a complex calculation for coherence and readability. (Capability 1, 2, 3, 4 and 5)
- Produce clearly reasoned proofs of general properties of Markov chains at an appropriate level of difficulty. (Capability 1, 2, 3, 4 and 5)
- Independently research and present a more advanced topic. (Capability 1, 2, 3, 4 and 5)
|Learning Outcome Addressed
Markov chains in discrete time: n-step transition probabilities, transition matrices. The Strong Markov Property. Recurrence and transience. Hitting probabilities. Stationary distributions and measures. Generating functions. Random walks. Branching processes. Markov chains in discrete space and continuous time: transition rates, infinitesimal generator, Q matrices, Kolmogorov equations. Resolvents and first hitting times. Equilibrium distributions. Time reversibility. Ergodic theorem.
This course is a standard 15 point course and students are expected to spend 150 hours overall in each 15 point course that they are enrolled in. Students are expected to spend 10 hours per week working on this course during each of the 12 teaching weeks, plus an additional 30 hours overall in preparation for tests/final examination (150 hours in total).
For this course, in each teaching week you can typically expect 3 hours of lectures, a 1 hour tutorial, 1-3 hours of reading and thinking about the content, and 3-5 hours of work on assignments and/or test preparation.
Participation is strongly recommended at scheduled activities including tutorials to successfully complete the course.
Lectures will be available as recordings. Other learning activities such as tutorials will not typically be available as recordings, but the course may additionally include live online tutorials. Online Q&A help will also be available via Piazza on the course Canvas site. All course materials will be made available online via Canvas.
Coursework may be submitted online via Canvas. Attendance on campus is not required for any mid-semester test, which will be online where applicable.
The activities for the course are scheduled as a standard weekly timetable.
Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).
Please remember that the recording of any class on a personal device requires the permission of the instructor.
Recommended further reading:
"Markov Chains" by J.R.Norris (Cambridge University Press)
"Probability: an introduction" by G.R.Grimmett and D. Welsh (Oxford University Press)
During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.
At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.
Your feedback helps to improve the course and its delivery for all students.
The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed against online source material using computerised detection mechanisms.
Class representatives are students tasked with representing student issues to departments, faculties, and the wider university. If you have a complaint about this course, please contact your class rep who will know how to raise it in the right channels. See your departmental noticeboard for contact details for your class reps.
The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.
You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.
All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.
Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website http://disability.auckland.ac.nz
If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.
If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page https://www.auckland.ac.nz/en/students/academic-information/exams-and-final-results/during-exams/aegrotat-and-compassionate-consideration.html.
This should be done as soon as possible and no later than seven days after the affected test or exam date.
In the event of an unexpected disruption, we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and course assessment continues to meet the principles of the University’s assessment policy. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator/director, and if disruption occurs you should refer to the university website for information about how to proceed.
The delivery mode may change depending on COVID restrictions. Any changes will be communicated through Canvas.
Student Charter and Responsibilities
The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html.
Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.
In this course students may be asked to submit coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. In exceptional circumstances changes to elements of this course may be necessary at short notice. Students enrolled in this course will be informed of any such changes and the reasons for them, as soon as possible, through Canvas.