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  "body": "\n## Definition\n\nLet $\\mu$ be a probability distribution on $\\{0, 1, 2, \\dots\\}$, the offspring distribution. Let $(\\xi_{n, i})_{n \\ge 0, \\, i \\ge 1}$ be [[i.i.d.|0AJN7WHBJZ]] random variables with law $\\mu$.\nThe Galton–Watson process with offspring distribution $\\mu$ is the [[stochastic process|0QH5ENKZM1]] $(Z_n)_{n \\ge 0}$ with\n$$Z_0 = 1 \\quad \\text{and} \\quad Z_{n+1} = \\sum_{i=1}^{Z_n} \\xi_{n, i} \\quad \\text{for } n \\ge 0.$$\nHere $Z_n$ is the size of generation $n$, and an empty sum is $0$.\n\n## Remarks\n\n$(Z_n)$ is a [[Markov chain|033TJBZDG0]] on $\\{0, 1, 2, \\dots\\}$, and $0$ is an absorbing state. Extinction is the event that $Z_n = 0$ for some $n$.\nLet $m$ be the mean of $\\mu$ and let $G$ be its [[probability-generating function|0Z8PSSAVBA]]. Then $E[Z_n] = m^n$, and the generating function of $Z_n$ is the $n$-fold composition $G \\circ \\dots \\circ G$.\nThe extinction probability is the smallest solution $q \\in [0, 1]$ of $G(q) = q$.\nAssume $\\mu(\\{1\\}) \u003c 1$. Then $q = 1$ if $m \\le 1$, and $q \u003c 1$ if $m \u003e 1$.\n\n\n\n## Explanations\n\n### L1 human\n\nThe Galton–Watson process follows a family tree in which each person has a random number of children.\nIf people have on average at most one child, the family dies out with probability $1$. If they have more, it can survive forever.\n",
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