Probability II
Lectures held on Tsinghua Campus, Teaching Building 4, room 4206, Thursday 9:50-11:25 and Friday 13:30-15:05, starting February 26 for 12 weeks.
Grading based on: Homework (15%), Lecture writeup (20%), Midterm (30%), Final Exam (35%).
Grading based on: Homework (15%), Lecture writeup (20%), Midterm (30%), Final Exam (35%).
Lecturer
Date
26th February ~ 27th June, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Thursday | 09:50 - 11:25 | Tsinghua Building 4, Room 4206 | ZOOM 02 | 518 868 7656 | BIMSA |
| Friday | 13:30 - 15:05 | Tsinghua Building 4, Room 4206 | ZOOM 02 | 518 868 7656 | BIMSA |
Prerequisite
Law of Large numbers, Central Limit Theorem, Borel-Cantelli Lemma, Conditional expectation (Kolmogorov definition), Definition of a martingale.
Syllabus
Topics Covered:
1. Applications of Martingales: 1.1 Branching processes: classification 1.2 Growth and structure of Galton-Watson trees 1.3 Polya Urns
2. Finite Markov chains: 2.1 Irreducibility, periodicity, reversibility 2.2 Stationary measure 2.3 Total variation distance 2.4 Convergence theorem 2.5 Mixing time
3. Markov chains on a countable state space: 3.1 Transience, recurrence and positive recurrence 3.2 Random walk on regular trees and lattices
4. Random walk on networks: 4.1 Voltage, flow 4.2 Effective resistance 4.3 Energy of a flow 4.4 infinite networks
5. Continuous time Markov processes: 5.1 Definitions 5.2 Poisson process on a line 5.3 Infinitesimal generator 5.4 Stationary distribution 5.5 Queuing
6. Spatial processes: 6.1 Percolation on a tree and a lattice 6.2 Poisson process in space 6.3 Voronoi tessellation 6.4 Buffon needle
7. Brownian motion (BM): 7.1 Gaussian vectors 7.2 Construction and continuity 7.3 Reflection principle 7.4. Strong Markov property 7.5 BM in d dimensions.
1. Applications of Martingales: 1.1 Branching processes: classification 1.2 Growth and structure of Galton-Watson trees 1.3 Polya Urns
2. Finite Markov chains: 2.1 Irreducibility, periodicity, reversibility 2.2 Stationary measure 2.3 Total variation distance 2.4 Convergence theorem 2.5 Mixing time
3. Markov chains on a countable state space: 3.1 Transience, recurrence and positive recurrence 3.2 Random walk on regular trees and lattices
4. Random walk on networks: 4.1 Voltage, flow 4.2 Effective resistance 4.3 Energy of a flow 4.4 infinite networks
5. Continuous time Markov processes: 5.1 Definitions 5.2 Poisson process on a line 5.3 Infinitesimal generator 5.4 Stationary distribution 5.5 Queuing
6. Spatial processes: 6.1 Percolation on a tree and a lattice 6.2 Poisson process in space 6.3 Voronoi tessellation 6.4 Buffon needle
7. Brownian motion (BM): 7.1 Gaussian vectors 7.2 Construction and continuity 7.3 Reflection principle 7.4. Strong Markov property 7.5 BM in d dimensions.
Reference
Williams: Probability with Martingales
Karlin and Taylor: A first course in stochastic processes
Levin and Peres: Markov chains and mixing times (Second edition, 2017)
Moerters and Peres: Brownian motion
Karlin and Taylor: A first course in stochastic processes
Levin and Peres: Markov chains and mixing times (Second edition, 2017)
Moerters and Peres: Brownian motion
Audience
Advanced Undergraduate
, Graduate
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Yuval Peres obtained his PhD in 1990 from the Hebrew University, Jerusalem. He was a postdoctoral fellow at Stanford and Yale, and was then a Professor of Mathematics and Statistics in Jerusalem and in Berkeley. Later, he was a Principal researcher at Microsoft. In 2023, he joined Beijing Institute of Mathematical Sciences and Applications. He has published more than 350 papers in most areas of probability theory, including random walks, Brownian motion, percolation, and random graphs. He has co-authored books on Markov chains, probability on graphs, game theory and Brownian motion, which can be found at https://www.yuval-peres-books.com. His presentations are available at https://yuval-peres-presentations.com. He is a recipient of the Rollo Davidson prize and the Loeve prize. He has mentored 21 PhD students including Elchanan Mossel (MIT, AMS fellow), Jian Ding (PKU, ICCM gold medal and Rollo Davidson prize), Balint Virag and Gabor Pete (Rollo Davidson prize). He was an invited speaker at the 2002 International Congress of Mathematicians in Beijing, at the 2008 European congress of Math, and at the 2017 Math Congress of the Americas. In 2016, he was elected to the US National Academy of Science.