Past exam of the mathematics course of the University of Cambridge / 2002 / ia / Paper 2 / 12F (Probability)

A branching process models reproduction through successive generations. In the Galton-Watson process used here, every individual independently produces a nonnegative integer number of offspring with a common offspring distribution; these reproduction counts are independent across individuals and generations. If $X_n$ is the generation size, thenwhere the ${\xi }_{n,j}$ have that distribution and the empty sum is zero. Thus zero is absorbing: once a generation is empty, every subsequent generation is empty.

Given $X_n=k$, the next size is a sum of $k$ independent reproduction counts. Multiplication of their probability generating functions gives $\mathbb{E}[s^{X_{n+1}}\mid X_n=k]=G(s{)}^k$. Averaging this conditional expectation provesWith one initial individual, $G_0(s)=s$ and $G_1(s)=G(s)$, so $G_n$ is the $n$-fold composition of $G$ with itself.

The binomial series givesThus $p_0=1-\alpha >0$ and, for $k\ge 1$,with the factors after $\beta$ omitted for $k=1$. The signs follow from $0<\beta <1$. As $s\uparrow 1$, the series has nonnegative terms and its sum increases to $G(1)=1$, so monotone convergence gives ${\sum }_{k\ge 0}{p}_k=1$. These are therefore genuine probabilities on the nonnegative integers, proving that $G$ is a probability generating function. Its expected value is infinite, since $G^{\prime }(s)=\alpha \beta (1-s{)}^{\beta -1}\to \infty$; finiteness of the mean is not required for this branching process.

Write $G_n(s)=1-A_n(1-s{)}^{{\beta }^n}$. The composition formula gives $A_{n+1}=A_n{\alpha }^{{\beta }^n}$, with $A_0=1$. Consequently $A_n={\alpha }^{1+\beta +\cdots +{\beta }^{n-1}}$, and summing the geometric series yields the explicit generation generating functionThis is the fractional-power offspring generating function and its iterates.

Evaluation of the probability generating function at zero selects the constant coefficient, soTo identify this limit with ultimate extinction, let $E_n=\{X_n=0\}$. The absorbing-zero property of the branching process gives $E_n\subseteq E_{n+1}$. Ultimate extinction means that some finite generation is empty, namely $E={\bigcup }_{n\ge 0}{E}_n$. Continuity of probability on increasing events therefore givesThis set argument is essential: it uses absorption, rather than inferring extinction merely from a numerical limit of generation probabilities.