Dissertação de Mestrado Acadêmico - Iago de Carvalho Abalada

Títle: Limit Theorems for Branching Processes

Abstract: This thesis consists of a rigorous and self-contained study of branching processes, with a focus on the Galton–Watson process. The work presents the basic tools used in the analysis of such stochastic processes and, in a second moment, is dedicated to establishing properties of processes in the supercritical, subcritical, and critical regimes through limit theorems that describe the asymptotic behavior of the population size. In the supercritical regime, the Kesten–Stigum theorem provides conditions for the almost sure limit of a martingale associated with the process to be positive, establishing population growth at an exponential rate. In the subcritical regime, the Heathcote–Seneta–Vere-Jones theorem determines under which conditions the probability of the population surviving for a long time decays exponentially. For the critical regime, we demonstrate Kolmogorov's estimate for the survival probability and Yaglom's limit law, which states that the distribution of the population size, conditioned on non-extinction and suitably normalized, converges in distribution to an exponential distribution.

Banca Examinadora: 
Orientador: Rangel Baldasso - PUC-Rio
Simon Griffiths - PUC-Rio
Daniel Ungaretti Borges- UFRJ
Guilherme Henrique de Paula Reis - UFF
Suplente: Rodrigo Bezerra de Matos - PUC-Rio

Data: 09 de abril de 2026
Horário: 11h
Sala: L856