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Probability II / Branching Processes

The Galton-Watson Process and Extinction Probability

The population size in a branching process is, itself, a Markov chain on {0,1,2,…}\{0,1,2,\ldots\} — but it is one where the exit-distribution machinery of the previous topic runs into a genuine obstacle: the state space is infinite, so the matrix method (I−r)−1(I-r)^{-1} has no direct analogue. This note introduces the tool built specifically to handle this: the probability generating function, which converts "compose many independent copies of a random family size" into "iterate a single function" — turning Galton's 1873 question ("what is the probability a family name dies out?") into a clean fixed-point equation.
Learning objectives. By the end of this note you should be able to:
  • set up the Galton-Watson process and verify it is a Markov chain;
  • state the generating-function composition rule for nn-generation sizes;
  • state the extinction-probability theorem and identify η\eta as the smallest non-negative root of s=G(s)s=G(s);
  • compute the extinction probability explicitly for the geometric offspring distribution.

In this lesson

  1. The Galton-Watson process
  2. The generating-function machinery
  3. The extinction probability

The full lesson, with worked examples and exercise solutions, is for members.

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