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}.

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