ECE 534: RANDOM PROCESSES, SPRING 2026

• 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:

• Yuan-Pon Chen, yuanpon2 at illinois dot edu
  Office Hours: TBA (starting the week of 26th of January, 2026).

• Peizhi Niu, peizhin2 at illinois dot edu
  Office Hours: None. Peizhi is tasked with preparing HW solutions.

Announcements:

• In observance of MLK Day, there will be no lectures on Monday, the 19th of January, 2026.
• Quiz: January 21st, 2026, in-class. 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. Midterm exam times: March 2026, TBA and April 2026, TBA. Both exams are in-class. 
  • The material covered in the exams will be based on Prof. Leveque's notes (up to page 43 and included), Prof. Hajek's text and supplementary material on random graphs, matrices and percolation theory. The material in the two foremost texts is pretty much identical, but Prof. Hajek's book has more examples and different proofs. 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.
  • The exams are closed-notes. No textbooks, No Internet resources, etc.
• HW1 will be made available on the 28th of January  here. Due Feb 9th. You can also find the LaTeX template here or in the course detail section below.
• HW2 will be made available on the 9th of February here. Due February 18th. You can also find the LaTeX template here or in the course detail section below.
• HW3 will be made available on the 18th of February here. Due February 27th. You can also find the LaTeX template here or in the course detail section below.
• HW4 will be made available on the 27th of February here. Due 10th of March. You can also find the LaTeX template here or in the course detail section below.
• HW5 will be made available on the 10th of March here. Due 24th of March. You can also find the LaTeX template here or in the course detail section below.
• HW6 will be made available on March 24th here. Due April 7th. You can also find the LaTeX template here or in the course detail section below.
• HW7 will be made available on April 7th here. Due April 21st. You can also find the LaTeX template here or in the course detail section below.
• Please use Gradescope to submit your homework. Please email the TAs 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.

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


• 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.
• 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.:

• There will be approximately 7-8 Homework assignments. The Homeworks will not be graded, but you will be provided with solutions.
   
• 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.
   
 
• Your course grade will be determined by your performance in the exams, M1, M2 and F, according to the formula:
  
  Score = 0.3MT1 +0.3 MT2+0.4F ,
  
  where H, MT1, MT2 and F are normalized to 100.