grandes-ecoles 2015 QIII.C.1

grandes-ecoles · France · centrale-maths1__psi Probability Generating Functions Branching process and extinction probability via PGF iteration
We consider the Galton-Watson process. We assume $m\leqslant 1$. We denote, for $n\in\mathbb{N}^*$, $Z_n=1+\sum_{i=1}^n Y_i$ and $Z=1+\sum_{n=1}^{+\infty}Y_n$. We admit that $Z$ is a random variable defined on $\bigcup_{k\in\mathbb{N}}\{Y_k=0\}$.
Show that $Z$ is defined on a set of probability 1. 
We consider the Galton-Watson process. We assume $m\leqslant 1$. We denote, for $n\in\mathbb{N}^*$, $Z_n=1+\sum_{i=1}^n Y_i$ and $Z=1+\sum_{n=1}^{+\infty}Y_n$. We admit that $Z$ is a random variable defined on $\bigcup_{k\in\mathbb{N}}\{Y_k=0\}$.

Show that $Z$ is defined on a set of probability 1.
Paper Questions
QI.A.1 QI.A.2 QI.A.3 QI.B QI.C QI.D.1 QI.D.2 QI.E.1 QI.E.2 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.A.5 QII.B QII.C.1 QII.C.2 QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.B.1 QIII.B.2 QIII.C.1 QIII.C.2 QIII.C.3 QIV.A QIV.B QIV.C QIV.D QIV.E QIV.F QIV.G QV.A QV.B.1 QV.B.2 QV.C.1 QV.C.2 QV.D.1 QV.D.2 QV.E QV.F