Tejas Iyer

Branching Processes

TU Berlin, Summer 2026. Lecturer: Tejas Iyer. 5 ECTS, taught in English.

Materials

Organisation

Lectures were held on Tuesdays 16:00–18:00 in MA 550 during the summer semester 2026. Assessment was by oral exam.

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

The course explores 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 and advanced topics, depending on progress and 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 email me if you have any doubts.

Learning outcomes

Beyond preparing students for research, the course develops advanced probabilistic reasoning and problem-solving skills applicable in data-driven fields, and the ability to communicate complex ideas. Students will be able to:

  • Explain and prove key results concerning branching processes, such as the probability of extinction, the Kesten–Stigum theorem and change-of-measure arguments.
  • Apply probabilistic tools such as martingales.
  • Reason about new problems related to branching processes.

Assessment

A 45-minute oral exam, which examines your ability to:

  1. Explain and prove key results in a one-on-one interview setting (approximately three quarters of the exam).
  2. Solve new problems using the tools developed in the course, by reasoning about problems not seen before (approximately one quarter of the exam).

Theses

Students interested in the topic may use this course as preparation for a Bachelor’s or Master’s thesis with me until June 2027. Wolfgang König is also available to supervise theses related to this topic.

Key references

  • K. B. Athreya and P. E. Ney, Branching Processes. Dover, 2004.
  • R. Lyons and Y. Peres, Probability on Trees and Networks. Cambridge University Press, 2017.