Lecturer: Tejas Iyer
Class format: Lectures every Tuesday 16:00 - 18:00, Room MA 550
Contact: After lectures, or email me - [first].[lastname]@wias-berlin.de
Language: English
Credits: 5 ECTS
Course description: Branching processes provide a natural model for the growth of populations over time, such as animal populations or the number of infected individuals in a viral epidemic. Simple probabilistic models allow us to understand key features of population dynamics, including exponential growth, extinction probabilities, and critical thresholds. By the end of the course, students will gain a deeper understanding of applied discrete probability and the phase transitions that arise in complex stochastic systems.
Syllabus
This course will explore the classical discrete-time Bienaymé–Galton–Watson process.
Core Topics:
Analysis of branching processes via probability generating functions: Extinction probabilities, conditional limit theorems, Kolmogorov's estimate and the Yaglom limit law.
Martingales associated with branching processes: Seneta-Heyde norming and the Kesten Stigum theorem.
Spine techniques and change of measure.
Flexible/ advanced topics (depending on class progress/interest):
Random trees, conditioned Bienaymé–Galton–Watson trees and scaling limits
Branching random walks and Crump-Mode-Jagers branching processes
Computational methods: simulations and modelling applications such as epidemiological spread.
Prerequisites:
Probability I and II
Please contact me by email if you have any doubts.
Learning outcomes:
Beyond preparing students for research, this course will develop skills in advanced probabilistic reasoning and problem solving skills applicable in various data-driven fields. Students will also develop important skills in communication of complex ideas.
More precisely, on satisfying the requirements of this course, students will be well-equipped to:
Explain and prove key results concerning branching processes (e.g. the probability of extinction, the Kesten-Stigum theorem, change of measure arguments)
Apply probabilistic tools such as martingales.
Reason about new problems related to branching processes.
Assessment: The assessment for this course consists of a 45 minute oral exam. This will examine your abilities to:
Explain and prove key results in a one-on-one interview setting (approximately 3/4 of the exam).
Solve new problems using the tools developed in the course, by reasoning in the interview setting about problem(s) not seen before (about 1/4 of the exam).
Research Opportunities:
Students interested in the topic may use this course as preparation for a Bachelor’s or Master’s thesis with me until June 2027. Professor Wolfgang König is also available to supervise Bachelor’s and Master’s theses related to this topic.
Resources
Links to external resources will be provided as the course progresses. Lecture notes from the previous year are also available (some material will differ this year).
Key References
K. B. Athreya and P. E. Ney, Branching Processes. Dover Books on Mathematics, 2004.
R. Lyons and Y. Peres, Probability on Trees and Networks. Cambridge University Press, 2017.