Theory of Probability

Course schedule

Here’s the course schedule, with topics and readings. And, here is a single page with all the notes.

Week 1

Stochastic processes and Markov chains

  • Markov chains: examples
  • stochastic processes, generally
  • construction of Markov chains and the Markov properties; Chapman-Kolmogorov equation
  • reading: Durrett, 5.1, 5.2
  • whiteboard: day 1 // day 2 // day 3
  • homework (due 4/7): Durrett 5.1.2, 5.1.3, 5.2.7, 5.2.8
Week 2

Markov chains - general behavior

  • recurrence and transience
  • stationary measures and reversibility
  • convergence to stationarity
  • whiteboard: day 4 // day 5 // day 6
  • reading: Durrett, 5.3, 5.5
  • homework (due 4/14): Durrett 5.3.2, 5.3.4, 5.5.2, 5.5.6
Week 3

Markov chains - asymptotics

  • asymptotic behavior of Markov chains
  • convergence, via coupling
  • tail sigma-field
  • whiteboard: day 7 // day 8 // day 9
  • reading: Durrett, 5.6, 5.7
  • homework (due 4/21): Durrett 5.6.1, 5.6.7, 5.7.1 and this problem
Week 4

Ergodic theorems

  • stationary sequences and measure-preserving transformations
  • Birkhoff’s ergodic theorem
  • Benford’s law
  • whiteboard: day 10 // day 11 // day 12
  • reading: Durrett, 6.1, 6.2
  • homework: none, due to midterm
Week 5

Brownian motion: introduction and construction

  • review from Markov chains
  • Lévy’s construction of Brownian motion
  • Midterm: Tuesday
  • whiteboard: day 13 // day 15
  • reading: Durrett, 7.1
Week 6

Brownian motion: continuity, Markov properties, stopping times

  • Brownian scaling and measure-preserving transformations
  • Hölder continuity, nowhere-differentiability
  • germ field, hitting zero
  • stopping times and right-continuity
  • strong Markov property
  • reading: Durrett, 7.1-7.3
  • whiteboard: day 16 // day 17 // day 18
  • homework (due 5/12): 7.1.1, 7.1.3, 7.1.6, 7.2.1, 7.3.2
Week 7

Brownian motion: path properties and optional sampling

  • the maximum process
  • the hitting time process
  • the reflection principle
  • the zero set
  • optional sampling
  • Brownian martingales
  • reading: Durrett, 7.4-7.5
  • whiteboard: day 19 // day 20 // day 21
  • homework (due 5/19): 7.4.1, 7.5.1, and this problem
Week 8

Brownian motion: Itô’s formula

  • quadratic variation
  • Itô’s formula
  • stochastic integration
  • reading: Durrett, 7.6
  • whiteboard: day 22 // day 23 // day 24
  • homework (due 5/26): 7.6.2, 7.6.3, and this problem
Week 9

Brownian motion: Donsker’s theorem; higher-dimensional Brownian motion

  • Skorokhod embedding
  • Donsker’s theorem
  • consequences for random walk
  • Brownian bridge
  • harmonic functions in $\mathbb{R}^d$
  • reading: Durrett, 8.1, 8.4, 9.1
  • whiteboard: day 25 // day 26 // day 27
  • homework (due 6/2): 8.1.2 and this problem
Week 10

Multidimensional Brownian motion and the heat equation

  • the maximum principle
  • recurrence and transience in $\mathbb{R}^d$
  • the heat equation
  • reading: Durrett, 9.1-9.3
  • whiteboard: day 28 // day 29
  • no homework
Week 11

Final exam

  • 2:45pm, Tuesday, June 8th