Branching

Branching processes are a type of stochastic process that models the growth of a population. The process starts with one individual, called the ancestor, and each individual can produce a random number of offspring. The number of offspring is typically modeled by a probability distribution. The process continues recursively, with each individual producing offspring, and the process stops when no individual has any offspring.

Throughout, we consider a offspring distribution of a non-negative integer-valued random variable $X$ with mean $\mu =\mathbb{E}[X]$ and PMF $(p_k{)}_{k=0}^{\infty }$.

Generating Function

The generating function of the branching process is

$$g(s)=\mathbb{E}[s^X]=\sum _{k=0}^{\infty }{p}_k{s}^k.$$

Similar to Moment Generating Function and Characteristic Function, the generating function uniquely determines the offspring distribution and thus the branching process.¹

Extinction Probability

Let $\eta$ be the probability that the branching process goes extinct. Then, $\eta$ is the smallest fixed point of the generating function, i.e., $g(\eta )=\eta$. This is because, the process goes extinct iff the subprocesses generated by the offsprings of the ancestor all go extinct. Since they are i.i.d. as the original process, we have

$$\eta ={\mathbb{E}}_k[{\eta }^k]=\sum _{k=0}^{\infty }{p}_k{\eta }^k=g(\eta ).$$

Let’s examine the graph of $g$ on $[0,1]$. We have $g(0)=p_0$, $g(1)=1$, $g^{\prime }(0)=p_1$, $g^{\prime }(1)=\mu$, and $g^{\prime \prime }(s)\ge 0$. In the supercritical regime $\mu >1$, at least for one $k\ge 2$ we have $p_k>0$, and thus $g^{\prime \prime }(s)>0$ on $(0,1]$.

There are two intersections of $g$ and the line $y=s$ when $\mu >1$. Let ${\eta }_i$ be the probability that $(i+1)$-th generation goes extinct. We have ${\eta }_0=p_0$, ${\eta }_1={\sum }_{k\ge 1}{p}_k{p}_0^k=g({\eta }_0)$, and in general ${\eta }_i=g({\eta }_{i-1})$. Thus, the extinction probability approaches the lower intersection from below (the zig-zag lines).

We summarize the scenarios in the following table

$\mu$	PMF	$\eta$
$<1$	any	$1$
$>1$	$p_0=0$	$0$
$>1$	$p_0>0$	$(0,1)$
$=1$	$p_1=1$	$0$
$=1$	$p_1<1$	$1$

Note that when $\mu =1$, $p_1<1$ is equivalent to $\mathrm{Var}(X)>0$.

Derived Random Variable

Number of Offsprings

Let $Z_i$ represents the number of $i$-th generation offsprings. Let $Z={\sum }_{i=0}^{\infty }$. The process goes extinct iff $Z<\infty$. We have

$$\mathbb{E}[Z_i]=\mathbb{E}[\mathbb{E}[Z_i|Z_{i-1}]]=\mu \mathbb{E}[Z_{i-1}]=\cdots ={\mu }^i.$$

Thus,

$$\mathbb{E}[Z]=\frac{1}{1-\mu }.$$

When $\mu <1$, the above expectation is finite, and thus $Z$ is finite with probability one.

Extinction time

Let $T$ be the number of generations until extinction, i.e., $T=\min \{i:Z_i=0\}$. Then

$$P(T>k)=P(Z_k>0)=P(Z_k\ge 1)\le \mathbb{E}[Z_k]={\mu }^k.$$

Footnotes

1. In fact, the generating function is equivalent to the MGF by setting $s=e^t$. ↩