Math 6010 • Linear Models

Let \(\{X_{i,n}:i,n\geq 1\}\) be i.i.d. random variables with mass function \(\{f(x):x=0,1,2,\ldots \}\). Fix \(z\in \{1,2,\ldots \}\). Define random variables \((Z_n)\) as follows: \[ Z_0=z \qquad \text {and}\qquad Z_n=\sum _{i=1}^{Z_{n-1}}X_{i,n} \qquad \text {for } n\geq 1. \] I.e. \(Z_n\) is a branching process with offspring distribution \(f\) and initial population \(z\). For \(s\in [0,1]\) let \(g(s)=\sum _{x=0}^{\infty }f(x)s^x\). Suppose that \(s_0\in (0,1)\) solves \(g(s)=s\). Show that then \(s_0^{Z_n}\) is a martingale in the filtration \(\mathcal {F}_n=\sigma (X_{i,m}:i\geq 1,m\leq n)\). Use this to conclude that \[ \mathbb {P}(Z_n=0\text { for some }n\geq 0)=s_0^z. \]