GWB Branching Processes – John Robin Inston

1 What is a GWB Branching Process?

The Galton-Watson-Bienaymé (GWB) Branching Process serves as the simplest stochastic model of population growth. To construct the process we think about a family tree with one founding member (progenitor).

A GWB branching process $\{Z_{n}\}$ is defined by \[ Z_{n}=\sum_{i=1}^{Z_{n-1}}Z_{n,i}, \] where $Z_{n,i}$ are i.i.d. random variables with PMF $\{p_{k}\}$ where $p_k=\mathbb{P}(Z_{{n,i}=k})$ for $k=1,2,\dots$and $Z_{0}=1$.

Intuitively, $Z_{n}$ describes the size of the family at generation $n$ and $Z_{n,i}$ describes the size of an individual branching at generation $n$. The process describes the population growth from a single progenitor when at each iteration, each individual has the same probability of having different numbers of children.

Some immediate observations include:

$Z_{n}=0\implies Z_{i}=0$ for all $i>n$.
$Z_{n,i}$’s and $Z_{n-1}$ are independent.
$Z_{1}=Z_{1,1}\stackrel{\mathcal{D}}=Z_{n,i}$ is the family size for one individual.

Since this is a discrete state SP we will often require the generating function to perform our analysis.

Consider the generator of the population size $Z_{n}$ denoted $G_{n}(s):=\mathbb{E}s^{Z_{n}}$. For the generator of $Z_{{n+m}}$ we have \[ G_{n+m}(s)=G_{m}(G_{n}(s))= \underbrace {G(G(G(\dots(G(s)) \dots)))}_{n+m} \] where $G(s):=\mathbb{E}s^{Z_{{n,i}}}=\sum_{k=0}^\infty p_{k}s^k$ is the generator of one branching.

Proof: We write $Z_{n+1}=Z_{n+1,1}+Z_{n+1,2}+\dots+Z_{n+1,Z_{n}}$. The RV $Z_{n+1}$has generator $G_{n+1}$ while $Z_{n+1,i}$ has generator $G$ for all $i$. From results on the generators of sums of independent summands - we have that \[ \begin{align} G_{n+1}(s)=G_{Z_{n}}(G(s))=G_{n}(G(s)). \end{align} \] $\square$

Let $\mu=\mathbb{E}Z_{1}$ and $\sigma^2=\operatorname{Var}(Z_{1})$ be the average size and variance of an individual family. Then, \[ \mathbb{E}Z_{n}=\mu^n \] and $$ (Z_{n})= \[\begin{cases}n\sigma^2&\text{if }\mu=1 \\ \sigma^2(\mu^n-1)\mu^{n-1}(\mu-1)^{-1}&\text{if }\mu \neq 1. ::: \end{cases}\]

Proof: From PGF results for factorial moments we recall that \[ \mathbb{E}Z_{n}=G_{n}'(1)\quad \& \quad\mathbb{E}Z_{n}(Z_{n}-1)=G_{n}''(1). \] From the result above we also have that $G_{n}(s)=G(G_{n-1}(s))$ so that \[ G_{n}'(s)=G'(G_{n-1}(s))\cdot G_{n-1}'(s). \] For $s=1$ we have that \[ G_{n-1}(1)=\sum_{k=0}^\infty 1^kp_{Z_{n-1}}(k)=1;\quad G'(1)=\mathbb{E}Z=\mu;\quad G_{n-1}'(1)=\mathbb{E}Z_{n-1}, \] from which we see that \[ \mathbb{E}Z_{n}=G_{n}'(1)=\mu \mathbb{E}Z_{n-1}=\mu(\mu \mathbb{E}Z_{n-2})=\dots=\mu^n. \] To find the variance we write in terms of factorial moments as \[ \operatorname{Var}(Z_{n})=\mathbb{E}Z_{n}^2-(\mathbb{E}Z_{n})^2=\mathbb{E}[Z_{n}(Z_{n}-1)]+\mathbb{E}Z_{n}-(\mathbb{E}Z_{n})^2=G_{n}''(1)+\mu^n-(\mu^n)^2. \] To complete the proof we compute $G_{n}''(s)$.

2 GWB Extinction Probability

The extinction of a branching process is the event that the random sequence $\{Z_{n}\}$ consists of zeros for all but a finite number of values of $n$. This can be written mathematically as \[ \{\text{extinction}\}=\{Z_{n}=0\text{ for some }n\}=\{Z_{1}=0\text{ or }Z_{2}=0\text{ or }\dots\}=\cup_{n=1}^\infty\{Z_{n}=0\}. \] We notice that $\{Z_{n}=0\}\subseteq \{Z_{n+1}=0\}\subseteq \dots$ since extinction at time $n$ implies the process will remain at $0$ for future generations. By the monotonicity of probability, there exists a limit \[ \lim_{ n \to \infty } \mathbb{P}(\{Z_{n}=0\})=\mathbb{P}(\cup_{n=1}^\infty\{Z_{n}=0\}). \] :::{.definition data-title=“GWB Probability of Extinction”}

The probability of extinction of a GWB process $\eta$ is given by \[ \eta:=\mathbb{P}(\cup_{n=1}^\infty\{Z_{n}=0\})=\lim_{ n \to \infty } \mathbb{P}(Z_{n}=0)=\lim_{ n \to \infty } G_{n}(0). \]

Considering the behavior of $\eta$ for different values $\mu$ leads to the following useful theorem.

Assuming $0<p_{0}=\mathbb{P}(Z_{1}=0)<1$, $\eta:= \lim_{ n \to \infty }\mathbb{P}(Z_{n}=0)$ is the smallest non-negative root of the equation $G(s)=s$. Furthermore: 1. If $\mu=\mathbb{E}Z_{1}=1$ and $\sigma^2=\operatorname{Var}(Z_{1})>0$, then $\eta=1$ (critical case) 2. If $\mu<1$, then $\eta =1$ (subcritical case) 3. If $\mu>1$, then $\eta<1$ (supercritical case)

Note that if $p_{0}=0$ then $\eta=0$ and if $p_{0}=1$ then $\eta=1$. Further, if $\mu>1$ then $\mathbb{E}Z_{n}=\mu^n\uparrow \infty$ so, on average, population size increases to infinity. Contrarily, if $\mu<1$ then $EZ_{n}=\mu^n\downarrow 0$ so, on average, population decreases to zero.

Considering the subcritical case we see that as $n\rightarrow \infty$ the process can either explode or become extinct

Assume $p_{1}=\mathbb{P}(Z_{n,j}=1)<1$. No matter what the finite value of $\mu=\mathbb{E}Z_{1}$ is, for any positive finite integer $k$ we have \[ \lim_{ n \to \infty } \mathbb{P}(Z_{n}=k)=0 \] for $k=1,2,\dots$. Moreover, there is a positive probability of explosion $\mathbb{P}(Z_{n}\rightarrow\infty)=1-\eta \in(0,1)$, and positive probability of extinction $\mathbb{P}(Z_{n}\rightarrow 0)=\eta \in(0,1)$.

Furthermore, we obtain the following result about the rate of growth to infinity for trees where $Z_{n}\rightarrow \infty$.

If $\mu>1$, $\mathbb{E}Z_{1}^2<\infty$, then there is an a.s. convergence of the random sequence \[ W_{n}:=\frac{Z_{n}}{\mu^n}\rightarrow W~a.s. \] where limit $W$ is a random variable with mean $\mathbb{E}W=1$ and $\operatorname{Var}(W)>0$, i.e. $W$ is a non-degenerate random variable that is not constant. Also, $W=0$ if and only if $Z_{n}\rightarrow 0$.