2.2020 - On multi-type branching processes Marc PEIGNE

Universite de Tours / University of Science, Ho Chi Minh City  ́

M2 master thesis in mathematics 2020/2021
On multi-type branching processes

Marc PEIGNE,  ́ (1)

Summary.
The purpose of this master thesis is to describe the limit theory of multi-type

branching processes. There is an extensive literature on branching processes on ge-
neral stat spaces which is outside the spirit and the scope of this thesis and we will

concentrate on population with a finite number of distinct particles.
There is a natural and well known classification of single type processes according
to their criticality condition. Their extinction probability appears as the smallest
fixed point of their generating function and the speed of convergence to extinction
has been studied during the course.
We will see that this classification is still valid in the multi-type case, which
requires a deep understanding of the Frobenius theorem. Extinction probability,
transience and limit theorems for the subcritical, critical and supercritical case will
be studied carefully.
References.
[1] Athreya K.B. & Ney P. E. (1972) Branching processes, Springer-Verlag
Berlin Heidelberg New York
[1] Harris T. E. (1963) The Theory of Branching Processes, Grundlehren der
mathematischen Wissenschaften, Springer-Verlag Berlin Heidelberg

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1. Marc PEIGNE, IDP, UMR 7013, Facult ́e des Sciences et Techniques, Parc de Grandmont,  ́
37200 Tours. mail : peigne@univ-tours.fr