Introduction

Tuesday

• Topic: Sections 1.1, 1.2: Introduction. We start to get familiar with the notion of a stochastic process and discuss the terms state space, index set, Markov chain, transition state diagram, and transition matrix.
• Class materials: Lecture notes, worksheet
• After class: Work on Homework 0.

Thursday

• Topic: Sections 2.1, 2.3: \(n\)-step transitions. We give a formal definition of a discrete time, discrete state Markov chain and begin to discuss how the transition matrix of a Markov chain can be used to understand multi-step transition probabilities using the conditional law of total probability.
• Class materials: Lecture notes, worksheet
• After class: Work on Homework 1 and make sure you can run RStudio in class Tuesday.

Chapter 2

Tuesday

• Topic: Sections 2.2, 2.3, 2.4: Random walks. We continue our discussion on using matrix powers to understand \(n\)-step transition probabilities. Our main example will be a random walk on a graph, but the ideas hold for all kinds of Markov chains.
• Class materials: Lecture notes, worksheet
• After class: Read about the distribution of \(X_n\) and joint distributions in Section 2.3.

Thursday

• Topic: Section 2.3: Joint distributions. We discuss the (unconditional) distribution of \(X_n\) and joint distributions of a Markov chain at multiple time steps.
• Class materials: Lecture notes, worksheet
• After class: Read about a random walk on a cycle in Section 2.4 and about simulation in Example 2.19 of Section 2.5.

Chapter 3

Tuesday

• Topic: Section 3.1: Limiting distributions. We begin our discussion of the notion of limiting distributions. Our aim in the chapter will be to understand the long term behavior of Markov chains theoretically, but we start with some numerical work.
• Class materials: Lecture notes, worksheet
• After class: Make sure to bring a laptop to class next time and download the rref.Rmd file in advance.

Thursday

• Topic: Section 3.2: Stationary distributions. We begin our discussion of how to turn the problem of understanding limiting distributions from a question about limits to a question about algebra. This entails introducing the idea of a stationary distribution and solving linear systems of equations.
• Class materials: Lecture notes, worksheet, worksheet solutions
• After class: Post a question on Piazza about Problem 3 in today’s worksheet or some other piece of class material so far. Feel free to respond to others’ questions. For Tuesday, read Examples 3.8 and 3.11 and the definition of communication class.

Chapter 3

Tuesday

• Topic: Section 3.3: Communication classes. We discuss what it means for states to communicate and get introduced to the ideas of recurrence and transience. This discussion, while separate from limiting and stationary distributions, is the first piece to developing an understanding of the Limit Theorem for Regular Matrices and more broadly the question of which Markov chains have a limiting distribution.
• Class materials: Lecture notes, worksheet
• After class: Read pages 96-98 on recurrence and transience.

Thursday

• Topic: Section 3.3: Recurrence and transience. Our aim is to deepen our understanding of recurrent and transient states. We discuss the related ideas of excursions, regeneration, and the geometric distribution in order to give a quantitative criterion for showing a state is recurrent or transient.
• Class materials: Lecture notes
• After class: Read Theorem 3.3, Corollary 3.4, and Lemma 3.5.

Chapter 3

Tuesday

• Topic: Section 3.3: Canonical decomposition. We discuss how the recurrence and transience series criterion, briefly introduced last time, is derived, and then show how it can be used to show that states in a communication class are either all recurrent or all transient.
• Class materials: Lecture notes, worksheet
• After class: Read about first step analysis in Section 3.4, focusing on Example 3.17.

Thursday

• Topic: Section 3.4: Irreducible Markov chains. We discuss finite state irreducible Markov chains and a theorem that states that these Markov chains always have a unique stationary distribution whose components can be found using the expected length of an excursion. We discuss a computational method called first step analysis for finding the expected length of an excursion.
• Class materials: Lecture notes
• After class: Read about periodicity in Section 3.5. Put your focus on Lemma 3.7 and Example 3.18.

Chapter 3

Tuesday

• Topic: Sections 3.5, 3.6: Periodicity and ergodicity. Through examples we have seen two main obstructions for the existence of a limiting distribution with positive components: multiple communication classes (which obstructs the limiting matrix from having equal rows) or states having a periodic structure (which prevents \(\lim_{n \to\infty} P^n\) from existing like in the random walk on the 6-cycle). We discuss periodicity in more depth and a new theorem, the Limit Theorem for Ergodic Markov chains, which tells us that once these obstructions are overcome we must have a limiting distribution: any finite state, irreducible Markov chain whose states have period 1 must have a limiting distribution with positive components.
• Class materials: Lecture notes, Ergodicity
• After class: Read the opening discussion in Section 3.8 up to Example 3.27.

Thursday

• Topic: Section 3.8: Absorbing chains. Our final aim in our study of discrete time Markov chains is to understand in detail the limiting behavior of a chain that starts in a transient state. We begin with the special case of absorbing Markov chains. These chains consist of transient states and states \(i\) which satisfy the property \(P_{ii} = 1\).
• Class materials: Lecture notes, Absorbing chains, R code
• After class: Read Theorem 3.11 and the second proof given.

Chapter 3

Tuesday

• Topic: Section 3.8: Absorbing chains. We discuss the expected number of steps until absorption. We also present the idea of using absorption probabilities to understand the limiting matrix of a Markov chain with transient states and two recurrent communication classes.
• Class materials: Lecture notes, More on absorption
• After class: Work on the exam!

Thursday

• Topic: Exam work day. Instead of a regular class, I’ll be available for office hours.
• After class: Work on the exam!

Chapter 6

Tuesday

• Topic: Section 6.1: Poisson process. We introduce our first example of a continuous time Markov chain, the Poisson process, which models the occurence of arrivals of events (for example, telephone calls or customers entering a store) over continuous time.
• Class materials: Lecture notes, Poisson process
• After class: Review your probability theory notes on the exponential distribution.

Thursday

• Topic: Section 6.2: Arrival times. We continue our study of the Poisson process, discussing the inter-arrival times in a Poisson process. We show that these times are exponentially distributed. The gamma distribution is also used to model the time of the \(n\)th arrival.
• Class materials: Lecture notes, Arrival times
• After class: Enjoy spring break!

Spring Break

Tuesday

• Topic: Spring break, no class.

Thursday

• Topic: Spring break, no class.

Chapter 6

Tuesday

• Topic: Section 6.4: Thinning and superposition. We spend the day discussing what kinds of statements and questions can be addressed about independent Poisson processes and independent exponentially distributed random variables.
• Class materials: Lecture notes, Thinning and superposition
• After class: Try the worksheet problems above, come ask questions about them, and read the introductory paragraphs of Sections 7.1 and 7.2.

Thursday

• Topic: Sections 6.3, 7.1: Introduction to continuous-time Markov chains. We introduce the definition of a Markov chain with finite or countable state space that evolves in continuous time. This involves the interrelated notions of hold times (the exponentially distributed amount time until a transition to another state occurs), the embedded chain (the discrete time Markov chain that determines which state to transition to), and the idea that in a certain sense a Poisson process can have at most one arrival in an infinitesimal time interval.
• Class materials: Lecture notes, Hold times and embedded chains
• After class: Read Example 7.5.

Chapter 7

Tuesday

• Topic: Section 7.2: Alarm clocks and transition rates. We work on more examples of continuous-time Markov chains and further discuss the idea that a CTMC can be defined by its transition rates.
• Class materials: Lecture notes, Transition rates
• After class: Read the introductory discussion of Section 7.3 and the proof of the Kolmogorov forward equation.

Thursday

• Topic: Section 7.3: Infinitesimal generator. We introduce the matrix \(Q\) of transition rates between states and its relationship with the transition probability function \(P(t)\). The Kolmogorov forward equation establishes a differential equation that relates these ideas, and the matrix exponential \(e^{tQ}\) enables us to solve the Kolmogorov forward equation.
• Class materials: Lecture notes, Infinitesimal generator
• After class: Read the Theorem 7.2 and the proposition and proof on the following page titled Stationary Distribution and Generator Matrix.

Chapter 7

Tuesday

• Topic: Community day, no class.

Thursday

• Topic: Section 7.4: Stationary distribution. We discuss limiting and stationary distributions in the regime of continuous time Markov chains. Our main goal is to establish an algebraic equation that can be solved to find stationary distributions analogous to the equation \(pi P=pi\) from the discrete time case.
• Class materials: Lecture notes, Limiting distribution
• After class: .

Project work

Tuesday

• Topic: Project work time.

Thursday

• Topic: Project work time.

Project work

Tuesday

• Topic: Project work time.

Thursday

• Topic: Project work time.

Presentations

Tuesday

• Topic: Project presentations.
• After class: Work on Exam 2. Enjoy your summer! Keep in touch!