= Galton-Watson process
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{title2=$Z_{n+1}=\sum_{j=1}^{Z_n}\xi_{n,j}$}
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A <Galton-Watson process> is a <branching process> that counts individuals by generation. Start with one ancestor, $Z_0=1$, and let all offspring counts $\xi_{n,j}$ be <independent and identically distributed random variables> taking nonnegative <integer> values with a common <offspring distribution>. Then
$$
Z_{n+1}=\sum_{j=1}^{Z_n}\xi_{n,j},\qquad \mathbb P(\xi_{n,j}=k)=p_k,\quad k\geq0.
$$
The <empty sum> is zero: zero is an <absorbing state>, so once a generation is empty every later generation is empty. The same <offspring distribution> applies to every individual in every generation. https://ocw.mit.edu/courses/18-445-introduction-to-stochastic-processes-spring-2015/27aaeb9eb9a9cecce93d9ad74413ddf5_MIT18_445S15_lecture19.pdf[Hao Wu, MIT 18.445, Lecture 19, slides 3–4]{ref}
For a concrete <offspring distribution>, take $p_0=1/4$, $p_1=1/4$, $p_2=1/2$, and $p_k=0$ for $k\geq3$. An individual has no children with <probability> $1/4$, one with <probability> $1/4$, and two with <probability> $1/2$. The offspring <expected value> is $\mu=5/4$. A possible sequence of generation sizes is $1,2,1,0,0,\ldots$: two children need not lead to a permanently growing family.
The <probability generating function> $F(s)=1/4+s/4+s^2/2$ gives the exact generation <probability distributions> through <branching-process generating-function iteration>. With $G_0(s)=s$ and $G_{n+1}(s)=F(G_n(s))$, the second generation has
$$
G_2(s)=\frac{11}{32}+\frac18s+\frac9{32}s^2+\frac18s^3+\frac18s^4.
$$
Thus $\mathbb P(Z_2=0)=11/32$ and $\mathbb E[Z_2]=25/16$. The figure shows full generation <probability mass functions>, computed by <polynomial> <function composition>, through generation six; no simulated frequencies are used.
\Image[/branching-generation-distributions.png]
{title=Exact Galton-Watson generation distributions: growing means and increasing extinction mass}
The contrast is striking: the <expected generation size in a Galton-Watson process> grows as $(5/4)^n$, while the <branching extinction probability> is $1/2$. By generation six, $\mathbb P(Z_6=0)\approx0.462186$ although $\mathbb E[Z_6]\approx3.814697$. Larger surviving families carry the rising <expected value>; that <mean> is an average over possible families, not a forecast of one family's path. The <Galton-Watson extinction fixed point> turns an unbounded number of future generations into a single equation for this <probability>.
The historical question was the disappearance of surnames passed through male descendants. In https://math.uchicago.edu/~shmuel/Modeling/Galton%20Watson.pdf[Watson and Galton, On the Probability of the Extinction of Families, pages 139–140], <Francis Galton> formulates a constant-law model counting sons who reach adulthood, and Watson computes later generation <probability distributions> by composing <generating functions>. Those are assumptions of their model, rather than a claim that actual human reproduction follows this <offspring distribution>.
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