**Definition (Stochastic process)** A stochastic process is a family of random variables $\{X(t), t\in \mathcal{I}\}$ on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, where $\mathcal{I}$ is a set called the index set.

Stochastic processes can take discrete or continuous values and can have a discrete or continuous index set $\mathcal{I}$. The branching process describes how a population evolves at discrete time intervals, and therefore takes discrete values and has a discrete index set $\mathcal{I}$.

**Definition (Branching process)** A branching process is a collection of random variables $X_0, X_1, X_2, ...$, representing a population of *nomads* which evolves as follows:
1. At $t = 0$ there is one nomad, $X_0 = 1$.
2. At every time-step, each nomad dies after giving birth to a *family* of new nomads of random size $C$.
3. The family sizes descending from each nomad are i.i.d. and are distributed according to the pmf $\mathbb{P}(C = k) = p_k, k = 0, 1, 2 ...$, called the *family-size distribution*.

## Probability generating functions We are interested in the distribution of the nomad population at future times. The recursive nature of the process gives a neat expression in terms of the generating function of the family-size distribution.

**Theorem (Probability generating functions of a branching process)** Let $X_t$ be a branching process with family-size distribution $\mathbb{P}(C = k) = p_k$, whose probability generating function is $G$. Then the probability generating functions $G_0, G_1, ...$ of $X_0, X_1, ...$ are
$$\begin{align}
G_0(s) = s, G_{t+1}(s) = G_t(G(s)), \text{ for } t = 1, 2, ... .
\end{align}$$

## Mean population

**Theorem (Mean population of a branching process)** The mean value of a branching process $X_t$ is
$$\begin{align}
\mathbb{E}(X_t) = \mu^t,
\end{align}$$
where $\mu = \mathbb{E}(C)$ is the mean family size. From this, it follows that as $t \to \infty$
$$\begin{align}
\mathbb{E}(X_t) \to \begin{cases}
0 & \text{ if } \mu < 1\\
1 & \text{ if } \mu = 1\\
\infty & \text{ if } \mu > 1
\end{cases}
\end{align}$$

This result is not surprising since each nomad will give birth to $\mu$ other nomads on average. It can be proved quickly using probability generating functions.

## Ultimate extinction We are also interested in the behaviour of the process as $t \to \infty$. If the population ever reaches $X_t = 0$, it will have become extinct.

**Definition (Extinction probability)** Given a branching process $X_t$, the extinction probability $e$ is
$$\begin{align}
e = \mathbb{P}(X_t = 0 \text{ for some } t \geq 0).
\end{align}$$
Alternatively, defining $E_t = \{X_t = 0\}$ as the event that the process is extinct by time $t$ and $e_t = \mathbb{P}(E_t)$ we have
$$\begin{align}
\{X_t = 0 \text{ for some } t \geq 0\} = \bigcup^\infty_{t = 0} E_t,
\end{align}$$
and since $E_t \subseteq E_{t+1}$ we have $e = \lim_{t \to \infty} e_t$ by the continuity of probability measures. Thus $e$ can equivalently be defined as the limit of $e_t$ as $t \to \infty$.

We are interested in the probability of ultimate extinction, that is the probability that $X_t = 0$ for some $t$. Again, generating functions give a neat answer for this quantity too.

**Theorem (Extinction probability theorem)** Let $X_t$ be a branching process, whose family sizes $C$ have common probability generating function $G$. The probability of extinction $e$ is the smallest non-negative root of the equation
$$\begin{align}
x = G(x).
\end{align}$$

The following theorem gives a necessary and sufficient condition for ultimate extinction to be certain.

**Theorem (Extinction survival theorem)** Let $X_t$ be a branching process whose family size distribution satisfies $\mathbb{P}(C = 1) \neq 1$. The probability of ultimate extinction satisfies $e = 1$ if and only if the mean family size satisfies $\mu \leq 1$.

From the definition of the probability of extinction, it follows that if $\mu \leq 0$, then $X_t \to 0$ in probability. Further, if the inequality is strict, $\mu < 1$, then $X_t \to 0$ in mean square - this does not hold for $\mu = 1$.

## References ```{bibliography} ./references.bib ```