STATS 721 : Foundations of Stochastic Processes


2021 Semester Two (1215) (15 POINTS)

Course Prescription

Fundamentals of stochastic processes. Topics include: generating functions, branching processes, Markov chains, and random walks.

Course Overview

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).

Course Requirements

Restriction: STATS 325

Capabilities Developed in this Course

Capability 1: Disciplinary Knowledge and Practice
Capability 2: Critical Thinking
Capability 3: Solution Seeking
Capability 4: Communication and Engagement
Capability 5: Independence and Integrity
Graduate Profile: Master of Science

Learning Outcomes

By the end of this course, students will be able to:
  1. 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)
  2. Distinguish between transient and recurrent states in given finite and infinite Markov chains. (Capability 1 and 3)
  3. Translate a concrete stochastic process into the corresponding Markov chain given by its transition probabilities or rates. (Capability 1, 2 and 3)
  4. Apply generating functions to identify important features of Markov chains. (Capability 1 and 3)
  5. Organise a complex calculation for coherence and readability. (Capability 1, 2, 3, 4 and 5)
  6. Produce clearly reasoned proofs of general properties of Markov chains at an appropriate level of difficulty. (Capability 1, 2, 3, 4 and 5)
  7. Independently research and present a more advanced topic. (Capability 1, 2, 3, 4 and 5)


Assessment Type Percentage Classification
Coursework 30% Individual Coursework
Final Exam 60% Individual Examination
Test 10% Individual Test
Assessment Type Learning Outcome Addressed
1 2 3 4 5 6 7
Final Exam

Key Topics

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.

Special Requirements

Attending tutorials to improve problem solving skills is highly recommended.

Workload Expectations

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.

For this course, in each teaching week you can typically expect 3 hours of lectures, a 1 hour tutorial, 2-3 hours of reading and thinking about the content, and 3-4 hours of work on assignments and/or test preparation.  Other weeks, including during the examination period, will typically be split among reading, thinking about the content and exam preparation.

Delivery Mode

Campus Experience

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 final exam will normally be on campus.
The activities for the course are scheduled as a standard weekly timetable.

Learning Resources

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)

Student Feedback

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.

In feedback from previous iterations of the course, students have said that they appreciate the weekly tutorials and the associated exercise sheets and quizzes, including the associated marks.  Students have emphasised that these course components help them master the skills taught in the course and are valuable preparation for the assignments and exam.  For this reason, this component of the course has been expanded over time and now contributes to the final mark.

Digital Resources

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.

Academic Integrity

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.


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.

Inclusive Learning

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

Special Circumstances

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

This should be done as soon as possible and no later than seven days after the affected test or exam date.

Learning Continuity

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 your assessment is fair, and not compromised. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator, and if disruption occurs you should refer to the University Website for information about how to proceed.

Level 1:  Delivered normally as specified in delivery mode.
Level 2: You will not be required to attend in person.  All teaching and assessment will have a remote option.
Level 3 / 4: All teaching activities and assessments are delivered remotely.

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


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 you may be asked to submit your 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. The final decision on the completion mode for a test or examination, and remote invigilation arrangements where applicable, will be advised to students at least 10 days prior to the scheduled date of the assessment, or in the case of an examination when the examination timetable is published.

Published on 28/01/2021 11:29 a.m.