# STATS 721 : Foundations of Stochastic Processes

## Science

### 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 STATS 125, STATS 210, STATS 225, or STATS 320) as well as some mathematics (one of MATHS 208, MATHS 250, MATHS 253, or equivalent).

This course will also provide good preparation for more advanced courses in probability (such as STATS 710, STATS 720, or STATS 723).

### Course Requirements

Prerequisite: 15 points from STATS 125, 210, 225, 320 with at least a B+ and 15 points from MATHS 208, 250, 253 Restriction: STATS 325

### Capabilities Developed in this Course

 Capability 3: Knowledge and Practice Capability 4: Critical Thinking Capability 5: Solution Seeking Capability 6: Communication Capability 7: Collaboration

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

### Assessments

Assessment Type Percentage Classification
Coursework 40% Individual Coursework
Final Exam 50% Individual Examination
Test 10% Individual Test
1 2 3 4 5 6 7
Coursework
Final Exam
Test

### Tuākana

Tuākana Science is a multi-faceted programme for Māori and Pacific students providing topic specific tutorials, one-on-one sessions, test and exam preparation and more. Explore your options at
https://www.auckland.ac.nz/en/science/study-with-us/pacific-in-our-faculty.html
https://www.auckland.ac.nz/en/science/study-with-us/maori-in-our-faculty.html

### Key Topics

Markov chains in discrete time:
•  n-step transition probabilities and 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:
• Q matrices, transition rates, and infinitesimal generator
• 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.

The coursework includes some guided independent learning of a more advanced topic and a group presentation.

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 examinations (150 hours in total).

For this course, a typical weekly workload includes:

• 3 hours of lectures
• A 1-hour tutorial
• 1-3 hours of reviewing the course content
• 3-5 hours of work on assignments and/or test preparation

### Delivery Mode

#### Campus Experience

Lectures will be available as recordings. Other learning activities such as tutorials will not typically be available as recordings. Online Q&A help will be available via Piazza on the course Canvas site. All course materials will be made available online via Canvas.

The activities for the course are scheduled as a standard weekly timetable. Participation in person is strongly recommended at scheduled activities including tutorials to successfully complete the course.

Attendance in person will be required for any on campus test and exam. Coursework will be submitted online via Canvas.

### Learning 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.

• "Markov Chains" by J.R.Norris (Cambridge University Press)
• "Probability: an introduction" by  G.R.Grimmett and D. Welsh (Oxford University Press)
• "Probability and Random Processes" by G.R.Grimmett and D.Stirzaker (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.

Some potential future improvements in the light of student feedback received:
• Trying to improve tutorial engagement and other ways of getting peer interaction.
• Consider giving a summary/revision lecture at end of course to connect together all topics learned (if time permits).

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 for potential plagiarism or other forms of academic misconduct, using computerised detection mechanisms.

### Class Representatives

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 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 .

### Disclaimer

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.

Published on 31/10/2023 10:54 a.m.