ECE 534: RANDOM PROCESSES, FALL 2026

• Class time and place: 10:00am-11:20pm WF, 1015 ECEB.
• Instructor: Prof. Olgica Milenkovic, milenkov at illinois dot edu.
• Office Hours: Wednesday 12:30-2:00pm in 311 CSL.

• Teaching Assistants:

• Yuan-Pon Chen, yuanpon2 at illinois dot edu
  Office Hours: Wednesday 4:00-5:00pm in 3034 ECEB.

Announcements:

• An exciting article by Terence Tao On AI and Mathematics.
• Course attendance is mandatory. You can nevertheless skip lectures if you have valid reasons (illness, personal issue, conference travel etc).
• Homework format and if to have homework at all will be discussed during the first lecture. Options include: HWs issued along with solutions, no grading; Recitations; AI-guided derivations with concept quizzes etc. The instructor is open to other suggestions as well.
• Quiz: February 16th, 2026, in-class. This quiz will test your background knowledge in undergraduate probability theory. The material to be covered includes the axioms of probability and sigma algebras, counting 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. The quiz questions from the Spring 2026 semester can be found here.
• The first midterm exam time and location: October 7th, 2026, from 6:30pm to 8:30pm, in Room TBA ECEB. Material to be covered will be announced two weeks before the exam.  
• The second midterm exam time and location: November 11th, 2026, from 6:30pm to 8:30pm, Room TBA. Material to be covered will be announced two weeks before the exam. 
• The final exam time and location: TBA  
• The exams are closed-notes. No textbooks, No Internet resources, etc.

• HW format, grading etc will be discussed during the first lecture. Given the rapid rise of AI we need to have a different approach to practice problems and learning. If we decide to have HWs, they will be posted in the following format: Here. Due TBA. Solutions.

Class Handouts and Supplements. We will not go over the whole list of subjects but select a subset according to class interest.

Course Details

• SUPERSET OF MATERIAL TO BE COVERED: Actual lecture content listed first


• Lecture 1 (August 26th), Topic: Review of ECE 313 (undergraduate probability) concepts.     
• -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
• Reading material:    Prof. Leveque's lecture notes I.     
• Reading material:    Prof. Leveque's lecture notes II.     
• Lecture 1, Topic 1, 2:    Peano's axioms.     
• 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.
• Lecture 22, Topic 1:    Random Graphs.
• Lecture 23, Topic 1:    Percolation Theory.
• 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.

  Additional References:
  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.:

• We will jointly decide during the first lecture how to organize HW submissions and if to use HWs as a means of leanring at all.
   
• There will be two mid-term exams and one final.
   
• 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.
   
 
• There will be no grading on the curve. Your course grade will be determined by your performance in the exams, MT1, MT2 and F, according to the formula:
  
  Score = 0.3MT1 +0.3 MT2+0.4F ,
  
  where MT1, MT2 and F are normalized to 100.