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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2011 • Question 8

Question

Math 6010 • Linear Models Spring 2011 • Question 8

Let \(Z_n\) be a Galton–Watson branching process with offspring distribution \(\{p_k,\ k=0,1,2,\ldots \}\) and \(Z_0=x\) (with \(x\) a positive integer), and let \[ f(\theta )=\sum p_k\theta ^k \] be the associated pgf. Suppose that \(\rho \in (0,1)\) satisfies \(f(\rho )=\rho \). Show that \(\rho ^{Z_n}\) is a martingale, and use this to conclude that \(\mathbb {P}(Z_n=0\text { for some }n\geq 0)=\rho ^x\).

Raw LaTeX

Let \(Z_n\) be a Galton--Watson branching process with offspring
distribution \(\{p_k,\ k=0,1,2,\ldots\}\) and \(Z_0=x\) (with \(x\) a positive
integer), and let
\[
  f(\theta)=\sum p_k\theta^k
\]
be the associated pgf. Suppose that \(\rho\in(0,1)\) satisfies \(f(\rho)=\rho\).
Show that \(\rho^{Z_n}\) is a martingale, and use this to conclude that
\(\Prob(Z_n=0\text{ for some }n\geq0)=\rho^x\).

Topic Tags

Estimator Distributions Martingales and Supermartingales Multivariate Normal Distributions Uniform Integrability and Truncation Methods