Cluster models

a.k.a. cascade distributions, Galton-Watson models

A type of count model for a Markov stochastic pure-birth branching process in discrete time. They can be generalised into non-Markov processes or into, say, or miscellaneous other types of branching process.

This needs a better intro, but the Galton-Watson process is the one here.

There are many standard expositions. Two good ones:

  • Gesine Reinert’s Introduction to Branching Processes: Parts 1 and 2.

  • Steven Lalley’s intro.

The distribution of subcritical processes are sometimes tedious to calculate, although we can get a nice form for the generating function for a geometric offspring distribution.

Set \(\frac{1}{\lambda+1}=p\) and \(q=1-p\). We write \(G^n\equiv G\cdot G\cdot \dots \cdot G\cdot G\) for the \(n\)-fold composition of \(G\). Then the (non-critical) geometric offspring distribution branching process obeys the identity

\[ 1-G^n(s;\lambda) = \frac{\lambda^n(\lambda-1)(1-s)}{\lambda(\lambda^n-1)(1-s)+\lambda-1} \]

This can get us a formula for the first two factorial moments, and hence the mean and variance.

More generally the machinery of Lagrangian distributions is all we need.

Maybe I should use (Dwass 1969) to get the moments? Dominic Yeo has a great explanation as always.

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