# Branching processes

Let ${\{Z_{n,j}: n\geqslant1, j\geqslant1\}}$ be iid non-negative integer valued random variables with ${\mathbb P(Z_{n,j} = k)=p_k}$. Define the Galton-Watson branching process ${\{Z_n: n\geqslant 0\}}$ by

$\displaystyle Z_0=1, Z_n = \sum_{j=1}^{Z_{n-1}}Z_{n,j}\quad (n\geqslant1).$

Let

$\displaystyle P(s) = \mathbb E[s^{Z_1}] \sum_{k=0}^\infty p_ks^k,\quad 0\leqslant s\leqslant 1.$

Let ${S=\sum_{n=0}^\infty Z_n}$ and ${Q(s) = \mathbb E[s^S]}$. Then

$\displaystyle Q(s) = \mathbb E[s^S] = \sum_{n=0}^\infty\mathbb E[s^S|Z_1=n]\mathbb P(Z_1=n).$

Since ${(S|Z_1=0)\stackrel{d}{=}1}$ and ${(S|Z_1=n)\stackrel d= n+nS}$ for ${n>0}$,

$\displaystyle Q(s) = p_0s + \sum_{n=1}^\infty p_n\mathbb E[s^{n+nS}] = p_0s + \sum_{n=1}^\infty p_n(sQ(s))^n = p_0(s-1) + P(sQ(s)).$

It follows that

$\displaystyle \mathbb E[S] = \lim_{s\uparrow1}Q'(1) = p_0 +\mathbb E[Z_1](\mathbb E[S] + 1),$

and hence that

$\displaystyle \mathbb E[S] = \frac{p_0+\mathbb E[Z_1]}{1-\mathbb E[Z_1]}$

(assuming ${\mathbb E[Z_1]<1}$).

For example, if ${P(s) = 1-p+ps}$, then

$\displaystyle Q(s) = (1-p)(s-1) + 1-p + psQ(s),$

and hence

$\displaystyle Q(s) = \frac{(1-p)s}{1-ps}.$

It follows that ${S}$ has a geometric distribution with parameter ${1-p}$, i.e.

$\displaystyle \mathbb P{S=n} = p^{n-1}(1-p), n=1,2,3,\ldots$

Moreover,

$\displaystyle \mathbb E[S] = \frac{1-p + p}{1-p} = \frac1{1-p}.$