Category Archives: Branching Processes

Covariance of a Branching Process

Show that for a branching process \( (Z_n)\) with expected offspring \(\mu\) one has

\[\mathbf{E}( Z_n Z_m)= \mu^{n-m} \mathbf{E}( Z_m^2)\]

for \(0\leq m\leq n\).

Basic Branching Process

Consider the branching process with offspring distribution given by \( P(X=0)=\alpha\), \( P(X=1)=\frac23-\alpha\) and \( P(X=2)=\frac13\) for some \(\alpha \in [0,\frac23]\).

  1. For what values of \(\alpha\) is the process certain to die out. ?
  2. For values where there is a probability of surviving forever, what is this probability as a function of \(\alpha\) ?

Geometric Branching Process

Consider a branching process with a geometric offspring distribution \( P(X=k) = (1-p)p^k\), for \(k=0,1,2,\dots\) . Show that the ultimate extinction is certain if \(p \leq \frac12\) and that the probability  of extinction is \((1-p)/p \) if \(p > \frac12\).

 

[Meester ex. 6.6.5]