# 632 Introduction to Stochastic Processes, Lecture 4

## Fall 2019

 Meetings: MWF 8:50-9:40 Van Vleck B239 Instructor: Timo Seppäläinen Office: 425 Van Vleck Phone: 263-3624 E-mail: seppalai and then at math dot wisc dot edu

This is the course homepage. Part of this information is repeated in the course syllabus that you find on Canvas. Here you find our weekly schedule and updates on scheduling matters. Deadlines from the Registrar's page.

632 is a survey of several important classes of stochastic processes: Markov chains in both discrete and continuous time, point processes, and renewal processes. The material is treated at a level that does not require measure theory. Consequently technical prerequisites for this course are light: calculus, introductory probability and linear algebra are sufficient. However, the material is sophisticated, so a degree of intellectual maturity and a willingness to work hard are required.

Good knowledge of undergraduate probability at the level of UW-Madison Math 431 (or an equivalent course) is required. This means familiarity with basic probability models, random variables and their probability mass functions and distributions, expectations, joint distributions, independence, conditional probabilities, the law of large numbers and the central limit theorem. If you wish to acquire a book for review, the Math 431 textbook Introduction to Probability by Anderson, Seppäläinen and Valkó is recommended.

In class we go through theory, examples to illuminate the theory, and techniques for solving problems. Homework exercises and exam problems are paper-and-pencil calculations with examples and special cases, together with short proofs.

A typical advanced math course follows a strict theorem-proof format. 632 is not of this type. Mathematical theory is discussed in a precise fashion but only some results can be rigorously proved in class. This is a consequence of time limitations and the desire to leave measure theory outside the scope of this course. Interested students can find the proofs in the literature. For a thoroughly rigorous probability course students should sign up for the graduate probability sequence Math/Stat 733-734 which requires a background in measure theory from Math 629 or 721. An undergraduate sequel to 632 in stochastic processes is Math 635 Introduction to Brownian motion and stochastic calculus.

### Textbook

Rick Durrett: Essentials of Stochastic Processes. 3rd edition. We expect to cover parts of Chapters 1-5. UW-Madison students can download this textbook for free through SpringerLink. Lecture notes will also be provided on Canvas.

### Canvas

Homework assignments, solutions to homework, and lecture notes will be posted on Canvas.

### Piazza

We will use Piazza for online class discussion. On Piazza you can post your math questions and answer other students' questions. You can get to Piazza through Canvas. Here is a link to our class Piazza page.

### Evaluation

Course grades will be based on homework (with occasional quizzes possible) (20%), two midterm exams (20%+20%), and a comprehensive final exam (40%). Midterm exams will be in the evenings on the following dates:

• Exam 1 Wednesday October 16 (Week 7) 7:15-8:45 PM Van Vleck B130
• Exam 2 Wednesday November 20 (Week 12) 7:15-8:45 PM Van Vleck B130
• Final exam: Thursday December 19, 10:05-12:05 PM Van Vleck B239.
To each exam you are allowed to bring handwritten notes: one 8x11 page to Exam 1, two pages to Exam 2, and three pages to the final exam. But no calculators, cell phones, or other gadgets will be permitted in exams and quizzes, only pencil and paper.

Here are grade lines that can be guaranteed in advance. A percentage score in the indicated range guarantees at least the letter grade next to it.

[100,90] A,   (90,87) AB,  [87,76) B,  [76,74) BC,  [74,62) C,  [62,50) D,  [50,0] F.

Course grades will not be curved, but depending the outcome of the exams the tentative grade lines above may be adjusted.

### Fall 2019 weekly schedule

Here we record class topics as we cover them.

Week Topics
1 9/4-6 Review of probability. Stochastic processes, state space, finite-dimensional probabilities. IID processes. Strong law of large numbers. Definition of a renewal process.
2 9/9-13 Law of large numbers for renewal and renewal-reward processes. Markov chains and transition probabilities. Computation with transition probabilities. Simple random walk, gambler's ruin, success runs. Multistep transition probabilities. Homework 0 due Wednesday.
3 9/16-20 Markov property into the infinite future. Probability of win in symmetric gambler's ruin. Strong Markov property. Recurrence and transience. Homework 1 due Wednesday.
4 9/23-27 Recurrence and transience. Simple random walk. Canonical decomposition of the state space. Absorption probabilities. Homework 2 due Friday.
5 9/30-10/4 Absorption probabilities. Invariant measures and distributions. Homework 3 due Friday.
6 10/7-11 Strong law of large numbers for Markov chains. Markov chain Monte Carlo (D. Anderson lectures).
7 10/14-18 Markov chain convergence theorem. Conditional expectations. Homework 4 due Monday. Exam 1 Wednesday evening 7:15-8:45.
8 10/21-25 Martingales. Optional stopping and the martingale limit theorem. Applications to gambling, Polya's urn and random walk.
9 10/28-11/1 Limiting ratio in Polya's urn. Probability generating function. Simple branching process. Homework 5 due Wednesday.
10 11/4-8 A martingale for the branching process. Properties of the exponential distribution. Poisson processes on the line. Homework 6 due Friday.
11 11/11-15 Poisson processes on the line. Age and residual life in the Poisson process and general renewal process. Homework 7 due Friday.
12 11/18-22 Exam 2 Wednesday evening 7:15-8:45. Begin continuous-time Markov chains.
13 11/25-27 Construction of continuous-time Markov chains. The meaning of jump rates. Jump rate as derivative of transition probability. Discrete-time MC run by a Poisson process. M/M/1 and M/M/s queues. Chapman-Kolmogorov equations. THANKSGIVING BREAK.
14 12/2-6 Kolmogorov's forward and backward equations. Matrix exponentials. Solution for a two-state process. Invariant and reversible distributions. Limit theorem for transition probabilities and the SLLN.
15 12/9-11 Review.

### Instructions for homework

• Homework is collected in class on the due date, or alternately can be brought to the instructor's office or mailbox by 1:20 PM on the due date. No late papers will be accepted. You can bring the homework earlier to the instructor's office or mailbox.
• Observe rules of academic integrity. Handing in plagiarized work, whether copied from a fellow student or off the web, is not acceptable. Plagiarism cases will lead to sanctions. You are certainly encouraged to discuss the problems with your fellow students, but in the end you must write up and hand in your own solutions.
• Organize your work neatly. Use proper English. Write in complete English or mathematical sentences. Answers should be simplified as much as possible. If the answer is a simple fraction or expression, a decimal answers from a calculator is not necessary. But for some exercises you need a calculator to get the final answer.
• As always in mathematics, numerical answers alone carry no credit. It's all in the reasoning you write down.
• Put problems in the correct order and staple your pages together. Do not use paper torn out of a binder.
• Be neat. There should not be text crossed out. Recopy your problems. Do not hand in your rough draft or first attempt. Papers that are messy, disorganized or unreadable cannot be graded.

### Interested in what is happening in probability theory at the research level?

Check out the Probability Seminar for talks on topics that might interest you.

Timo Seppalainen