
Math 547  Stochastic Processes
201718



Instructor

Louigi AddarioBerry 
louigi.addario@mcgill.ca 
Tel: (514) 3983831 (office) 
1219 Burnside Hall

Time and Location

TuesdayThursday, 8:3010:00, Burnside 1B24.

Office Hours

Hannah Cairns 
ahc238 atsymbol cornell dotsymbol edu
Office Hours: 1120 Burnside Hall, Monday 34, Friday 23.


Louigi AddarioBerry  Tuesday 1011, Thursday 1011

Course book

Grimmett and Stirzaker, Probability and Random Processes.
Additional material may be posted, either here or in the weekbyweek listing of topics, below.

Outline

We will cover a variety of topics related to stochastic processes, roughly covering Chapters 5.15.5, 6.16.4, 6.86.9, 6.136.14, 8, 9 and 10 of Grimmett and Stirzaker. This covers the following topics:
 Generating functions and their applications;
 Onedimensional random walk;
 Branching processes and agedependent bra©nching processes;
 Markov chains (transience and recurrence; classification of states; stationary distributions; ergodic theorem for markov processes; birth and death processes);
 Continuoustime Markov chains; Poisson processes in one and more dimensions;
 Reversible Markov chains and the electrical network interpretation;
 Mixing time basics, lower and upper bounding techniques;
 Stationary processes and ergodic theorems
 Renewal processes and the renewal theorem
Time permitting, we may also cover some of the following topics: random graph processes; the PerronFrobenius theorem; Brownian motion and Lévy processes; models from physics (percolation, Ising model); MCMC.

Notes


Final exam topics


Generating functions (general theory, applications to stochastic processes)

Random walk (biased SRW transience/recurrence; hitting times/first passage times; transience and recurrence in higher dimensions; RW on graphs (regular and lazy); connection with electrical networks)

Branching processes (fundamental theorem; specific examples; agedependent processes)

Markov chains (Definitions; how to use the Markov property; classification of states and chains; period; transience/recurrence; hitting and return times; stationary distribution; convergence to stationarity; one step calculations for probabilities and expectations)

Continuous time chains (all the same stuff as for discrete time, plus: explosion; birth processes; jump chains; forward and backward equations)

Reversible chains (equivalence with random walk on weighted graphs; electrical interpretation; resistance and energy; Thompson's principle)

Mixing time (total variation distance; couplings of Markov chains; mixing time bounds; exponential convergence to stationarity)

Poisson processes (definition in one and higher dimensions; mapping; superposition; thinning/colouring; conditioning; examples; Rényi's theorem)

Assignments


Prerequisites

MATH 356 and either MATH 247 or MATH 251. Taking Math 587 concurrently is a good idea if you haven't already taken it but is not required.

Evaluation

There will be 2 inclass midterms each worth 15% of the final grade. There will be 5 assignments; worth a combined 20% of the final grade. There will be a finalexam worth the remaining 50% of the final grade.
Assignments are to be submitted electronically to the instructor by the time indicated on the assignment, using McGill myCourses.

Language of submission

Conformément à la Charte des droits de l'étudiant de l'Université McGill, chaque étudiant a le droit de soumettre en français ou en anglais tout travail écrit devant être noté (sauf dans le cas des cours dont l'un des objets est la maîtrise d'une langue).
In accord with McGill University's Charter of Students' Rights, students in this course have the right to submit in English or in French any written work that is to be graded. This does not apply to courses in which acquiring proficiency in a language is one of the objectives.

Academic Integrity

McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures” (see www.mcgill.ca/students/srr/honest/ for more information).
