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 \(0\leq m\leq n\).
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 \(0\leq m\leq n\).
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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 \(\alpha \in [0,\frac23]\).
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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]
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Posted in Basic Stochastic Processes, Branching Processes, Exponential Random Variables
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