Math 547 - Stochastic Processes


2017-18





Instructor  


Louigi Addario-Berry
| louigi.addario@mcgill.ca | Tel: (514) 398-3831 (office) | 1219 Burnside Hall


Time and Location  


Tuesday-Thursday, 8:30-10:00, Burnside 1B24.


Office Hours  


Hannah Cairns
| ahc238 atsymbol cornell dotsymbol edu
Office Hours: 1120 Burnside Hall, Monday 3-4, Friday 2-3.


Louigi Addario-Berry | Tuesday 10-11, Thursday 10-11


Course book  


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


Outline  


We will cover a variety of topics related to stochastic processes, roughly covering Chapters 5.1-5.5, 6.1-6.4, 6.8-6.9, 6.13-6.14, 8, 9 and 10 of Grimmett and Stirzaker. This covers the following topics:

  • Generating functions and their applications;
  • One-dimensional random walk;
  • Branching processes and age-dependent bra©nching processes;
  • Markov chains (transience and recurrence; classification of states; stationary distributions; ergodic theorem for markov processes; birth and death processes);
  • Continuous-time 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 Perron-Frobenius 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; age-dependent 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 in-class 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 final-exam 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).