ECE 534: RANDOM PROCESSES, SPRING 2022

• Class time and place: 11:00am-12:20pm MW, 1015 ECEB
• Instructor: Prof. Olgica Milenkovic, milenkov at illinois dot edu
• Office Hours: Mondays 1:30-3:00pm in 311 CSL

• Teaching Assistants:

• Chao Pan, chaopan2 at illinois dot edu
  Office Hours: Wednesdays, 3:00-4:00pm (starting the week of the 24th of January, 2022). 3081 ECEB.

• Eli Chien, ichien3 at illinois dot edu
  Office Hours: Thursdays, 5:00-6:00pm (starting the week of the 24th of January, 2022). Zoom links will be shared by the CLASSmail.

Announcements

• Midterm exam times: March 21st, 7-9pm, ECEB 2017 and April 19th, 7-9pm, ECEB 2017. There will be both an in-person and Zoom sections for exams.  PLEASE NOTE THE CHANGE IN THE DATE FOR MIDTERM 1.
  • The instructions for the exam are here.
  • The material covered includes all of Prof. Leveque's notes (up to page 43 and included) and the first two Chapters of Prof. Hajek's text. The material in these two texts is pretty much identical but Prof. Hajek's book has more examples and some proofs that differ from Prof. Hajek's. You are not responsible for the material pertaining to Cauchy's criterion for convergence. Please make sure to work out the examples from the text.
  • You can bring only one side of a US Letter paper for notes. No textbooks, No Internet resources, etc.
  • Zoom meeting id for Midterm 1: 839 4648 2594. Password is sent via email.                                                                                                                                                        
• A conflict midterm exam will be granted only if (other events do not constitute conflicts)
  • you are taking the course online and are in a time zone for which 7-9pm CT falls in the middle of the night or some other inconvenient time. Please email me and Chao by the 25th of February. We will try to find a time that works for most and schedule a zoom-proctored exam.
  • you have another exam scheduled for that time. Please contact us as well with the course number and time for the exam that is causing the conflict, again by February 25th. We will again try to accommodate your request.
  • you have an emergency or health problems, you will not know about the conflict in advance. In that case, you will not be asked to take a conflict exam but will be given the same number of points you got on the second midterm.                                                                                                              
• HW1 is available at  here. Due Feb 18 11am. You can also find the LaTeX template here or in the course detail section below. All questions in Problem 5 are taken from the textbook.
• HW2 is available at  here. Due Mar 3 11am. You can also find the LaTeX template here or in the course detail section below. All questions in Problem 1 are taken from the textbook.
• HW3 is available at  here. Due Mar 23 11am. You can also find the LaTeX template here or in the course detail section below.
• HW4 is available at  here. Due April 15 11am. You can also find the LaTeX template here or in the course detail section below.
• HW5 is available at  here. Due May 4 11am. You can also find the LaTeX template here or in the course detail section below.
• HW6 is available at  here. Due May 11 11am. You can also find the LaTeX template here or in the course detail section below.
• There will be weekly recitations by TAs: 3:00-4:00pm, Thursday, in room 3081 ECEB.
• Please use Gradescope to submit your homework no matter which section (in-person or online) you are enrolled in. Please email TA Chao Pan if you did not receive the instructor's email about the entry code. We accept both scanned handwritten homework and typed homework in Latex. We will provide Latex templates for each homework for those interested.
• Echo360 recordings have been scheduled and Engineering's Echo360 services can be accessed here.
• To login, you will need to enter their University of Illinois email address. You’ll then be prompted to authenticate using their Active Directory (AD) password. Once authenticated, you should see a dashboard page displaying content for which they are listed as the instructor or enrolled as a student. Student Echo360 user accounts will be auto-generated based on enrollment rosters within 24 hours.
  Further documentation on how to use the Echo360 platform for students can be found here.
• For the lecture on Jan 24, if you find the resolution of the Echo360 recording to be too low, we also have a Zoom recording here and the material is listed in Course Details (Lecture 1, topic 1-5). We will set up the devices to make sure that future Echo360 recordings have the best quality.
• For the lecture on Feb 7, we met some technical issues with Echo360 so the camera did not move at the end. An extra recording for the last part of this lecture is provided here.

Class Handouts and Supplements from Fall 2018 (We may not go over this whole list in 2022)

Course Details

• In observance of MLK Day, there will be no lectures on Monday, the 17th of January, 2022.
• The lecture on the 19th of January will be held over Zoom, with the link shared in the CLASSmail. The Zoom recording link can be found here.
• The lecture, office hour on Feb 2 and recitation on Feb 3 will be through Zoom. Zoom link has been shared by the CLASSmail.
• The lecture on the 2nd of February will be held over Zoom, with the link shared in the CLASSmail. The Zoom recording link can be found here; the material is also available here. 
• Recitations will start on January 27th, 2022. Chao and Eli will solve problems related to the topics covered in class.
  • Jan 27 recitation material
  • Feb 3 recitation material, Zoom recording
  • Feb 10 recitation material, Zoom recording
  • Feb 24 material shared via email, Zoom recording
  • Mar 3 material shared via email, extra recitation material
  • Mar 10 material shared via email
  • Apr 28 recitation material
• There will be an offline Background Material Quiz. The quiz will cover material in basic probability theory, including the axioms of probability, combinatorial approaches to computing probability, conditional probability, independence of events, Bayes' formula, random variables, CDFs, PMFs and pdfs, expectations of random variables, functions of random variables and sigma-algebras.
• Grading policy for the homework: The TA will randomly select half of the problems for grading and your score will be solely based on your solutions for the selected problems. Please make sure to attempt to solve every problem! Template for LaTeX user can be found here.
• There will be one quiz, two midterms and one final exam.
  • Midterm exam times: March 21st, 7-9pm, ECEB 2017 and April 19th, 7-9pm, ECEB 2017.
  
  
  • MATERIAL FROM 2018: PLEASE NOTE THAT NEW HOMEWORKS AND HANDOUTS WILL BE POSTED ABOVE
  
  
  • Reading material:    Prof. Leveque's lecture notes I.     
  • Reading material:    Prof. Leveque's lecture notes II.     
  • Lecture 1, Topic 1:    Peano axioms.     
  • Lecture 1, Topic 2:    Godel's Incompleteness Theorems.     
  • Lecture 1, Topic 3:    Cantor's theorem.     
  • Lecture 1, Topic 4:    Aleph numbers and the Continuum hypothesis.     
  • Lecture 1, Topic 5:    Cardinality of the Borel sigma-algebra.  
  • Lecture 2, Topic 1:    Kolmogorov's axioms: Prof. Hajek's text, Chapter 1.  
  • Lecture 2, Topic 2:    Continuity of probability, Prof. Hajek's book, Page 3.
  • Lecture 2, Topic 3:    Length as a measure and Vitali sets: Prof. Hajek's text, Problem 1.37.    
  • Lecture 3, Topic 1:    The Cantor set.     
  • Lecture 3, Topic 2:    Cylinder sets.  
  • Lecture 3, Topic 3:    Product Measures, Example from Prof. Hajek's book, Page 3.  
  • Quiz Review Session:    Spring 2017 Quiz. ; Solutions.  
  • Lecture 4, Topic 1:    Random variables and Measurable Functions.  
  • Lecture 4, Topic 2:    Cumulative distribution functions, Prof. Hajek's book, Page 9.  
  • Lecture 4, Topic 3:    Outer measures. Lebesgue measure.  
  • Lecture 5, Topic 1:    Continuity, Uniform continuity, absolute continuity; pdf's.  
  • Lecture 5, Topic 2:    Cantor function.  
  • Lecture 5, Topic 3:    Riemann and Lebesgue integrals.  
  • Lecture notes - advanced probability and measure theory (A. Dembo).  
  • Lecture notes - elementary probability (C. Grinstead).  
  • Lecture notes - advanced calculus (J. Muldowney).  
  • Lecture 6, Topic 1:    Dirichlet function.  
  • Lecture 6, Topic 2:    Covariance, correlation. Prof. Hajek's text, pg. 27-28.  
  • Lecture 6, Topic 3:    Moments of RVs, moment generating and characteristic functions, Prof. Hajek's book, 16-20.  
  • Lecture 6, Topic 4:    Characteristic functions always exist.  
  • Lecture 7, Topic 1:    Holder's inequality.  
  • Lecture 7, Topic 2:    Minkowski's inequality.  
  • Lecture 8, Topic 1:    limsup, liminf.  
  • Lecture 8, Topic 2:    limsup, liminf of sets.  
  • Lecture 8, Topic 3:    Borel-Cantelli Lemma, Prof. Hajek's book, Page 7.  
  • Lecture 9, Topic 1:    Almost sure convergence and convergence in probability. Prof. Hajek's text, pg. 40-42.  
  • Lecture 9, Topic 2:    Understanding the difference between almost sure and convergence in probability.  
  • Lecture 9, Topic 3:    Almost sure convergence: limsups and Borel-Cantelli lemma application.  
  • Lecture 9, Topic 4:    Relationships between various modes of convergence - proofs. See also Prof. Hajek's book, pages 40-52.  
  • Lecture 9, Topic 5:    Markov's inequality.  
  • Lecture 10, Topic 1:    Jensen's inequality, Prof. Hayek, pg. 58.  
  • Lecture 10, Topic 2:    Alternative characterizations of convergence in distribution (Prof. Hayek, pg. 46, Portmanteau lemma).  
  • Lecture 10, Topic 3:    Examples of convergence in distribution: Extremal order statistics.  
  • Lecture 11, Topic 1:    LLNs (Prof. Hayek, pp. 56-58).  
  • Lecture 11, Topic 2:    Terry Tao's blog on the strong LLNs.  
  • Lecture 12, Topic 1:    Central limit theorem (CLT), Prof. Hayek's text.
  • Lecture 12, Topic 2:    Lindeberg's CLT.
  • Lecture 12, Topic 3:    Berry-Essen theorem.
  • Lecture 13, Topic 1:    Chernoff bound and large deviations, Prof. Hajek's text, page 60.
  • Lecture 13, Topic 2:    Cramer's theorem, Prof. Hajek's text, page 61.
  • Lecture 13, Topic 3:    Chernoff bound for a sum of independent, nonidentically distributed Bernoulli variables.
  • Lecture 14, Topic 1:    Hoeffding and McDiarmid's inequality.
  • Lecture 14, Topic 2:    Martingales, Doob Martingales and Azuma's inequality.
  • Lecture 15, Topic 1:    Talagrand's inequality.
  • Lecture 15, Topic 2:    Random Processes, Chapter 4, Prof. Hajek's text: Definitions, Second order processes, mean and correlation/covariance function. Gaussian processes.
  • Lecture 15, Topic 2:    Random Processes, Chapter 4, Prof. Hajek's text: Gambler's ruin problem.
  • Lecture 16, Topic 1:    Polya's urn model.
  • Lecture 16, Topic 2:    Galton-Watson process.
  • Lecture 16, Topic 1:    Polya's urn model.
  • Lecture 17, Topic 1:    Random Processes, Chapter 4, Prof. Hajek's text: Brownian motion, Poisson processes, Martingale processes, Stationarity.
  • Lecture 17, Topic 2:    Markov Chains.
  • Lecture 18, Topic 1:    Perron-Frobenius Theory.
  • Lecture 18, Topic 2:    PageRank.
  • Lecture 19, Topic 1:    Random vectors, minimum mean square error estimation: Chapter 3, Prof. Hajek's text.
  • Lecture 19, Topic 2:    Chapter 3, Prof. Hajek's text: Innovation sequences.
  • Lecture 20, Topic 1:    How Kalman filtering works.
  • Lecture 21, Topic 1:    Random Matrices.
  • Syllabus: This is a graduate-level course on random (stochastic) processes, which builds on a first-level (undergraduate) course on probability theory, such as ECE 313. It covers the basic concepts of random processes at a fairly rigorous level, and also discusses applications to communications, signal processing, control systems engineering, and computer science. To follow the course, in addition to basic notions of probability theory, students are expected to have some familiarity with the basic notions of sets, sequences, convergence, linear algebra, linear systems, and Fourier transforms.

    Topics to be covered include:

    • Probability and Measure Theory Review;
    • Convergence of a Sequence of Random Variables;
    • Minimum Mean Squared Error Estimation;
    • Jointly Gaussian random variables and vectors;
    • Random Walks and Brownian Motion;
    • Discrete-Time Markov Chains;
    • Continuous-Time Markov Chains and Poisson Processes;
    • Stationarity, WSS and Ergodicity;
    • Martingales and the Azuma-Hoeffding Inequality;
    • Random Processes Through Linear Systems;
    • Kalman and Wiener Filtering;
    • Other topics as time permits.

     

    Required Text: B. Hajek, Random Processes for Engineers, Cambridge 2015. A soft copy of the book can be downloaded here. Preprints are also available through the ECE Store. I will also make frequent use of other texts, for example The Theory of Probability, by Santosh S. Venkatesh. See the Grainger Library reserves below.

    References in Grainger Library:
    Papoulis, A.; Probability, Random Variables, and Stochastic Processes
    Wong, Eugene; Introduction to Random Processes
    Wong, E./Hajek, B.; Stochastic Processes in Engineering Systems
    Stark, H./Woods, J.; Probability, Random Processes, and Estimation Theory
    Feller, W.M.; An Introduction to Probability Theory and its Applications
    Rudin, Walter; Principles of Mathematical Analysis
    Grimmett, G. R./Stirzaker, David R.; Probability and Random Processes
    Gallager; Discrete Stochastic Processes Santos Venkatesh, The Theory of Probability

    Exams, homework, grading, etc.:

  • There will be approximately 7 Homework assignments. Collaboration is permitted, however each student must write and submit independent solutions. Homework is due within the first 5 minutes of the class period on the due date (usually Wednesdays, unless otherwise stated). No late homework will be accepted (unless an extension is granted in advance by the instructor).
     
  • There will be two mid-term exams and one final. The dates will be announced shortly.
     
  • Exam conflicts: You have the responsibility to contact the instructor and TA at least one week before the exam if you have any acceptable conflicts, such as another exam, a documented job interview etc. Not being prepared for the exam does not constitute a conflict, so please do not request alternative exam dates for this reason.
     
  • You may bring one sheet of notes to the first hour exam, two to the second hour exam, and three to the final exam. You may use both sides of the sheets, the sheets are to be standard US (letter paper) or European (A4) size with font size 10 or larger printing (or similar handwriting size). The examinations are closed book otherwise. Calculators, laptop computers, tables of integrals, etc. are not permitted.
     
  • Your course grade will be determined by your performance in the H, M1, M2 and F, according to the formula:
    
    Score = 0.15H + 0.25MT1 +0.25 MT2+0.35F ,
    
    where H, MT1, MT2 and F are normalized to 100.