Branching Process

A branching process \(Z_{n}\) with reproduction law \(X\) counts the number of individuals at each generation

\[Z_n = \sum_{i=1}^{Z_{n-1}} X_{i},\]

where each \(X_i\) are independent with same distribution as \(X\) and \(Z_{0} = 1\).

Proposition. Denote \(\mu = EX\), \(\sigma^2 = \mbox{Var}(X)\), and \(\eta = P(\text{eventual extinction})\). Then,

1. \(\eta\) is the smallest non-negative root of \(G(s) = s\)
2. If \(\mu < 1\), then \(\eta = 1\)
3. If \(\mu > 1\), then \(\eta < 1\)
4. If \(\mu = 1\), then
   • If \(\sigma^2 = 0\), then \(\eta = 0\) since \(Z_{n} = 1\) is constant
   • If \(\sigma^2 > 0\), then \(\eta = 1\)

Note: if it’s given that the number of individuals at any generation is \(k\), then the probability of extinction is \(\eta^{k}\) as each individual can be thought as independent branching processes, so the probability of extinction is the probability that each process is eventually goes extinct.

Mean and Variance of the Branching Process

Let \(Z_{n}\) be a branching process with reproduction law \(X\) and let \(\mu = EX\), \(\sigma^2 = \mbox{Var}(X)\). The expectation of \(Z_{n}\) is

\[E[Z_{n}] = E[E[Z_{n} \mid Z_{n-1}]] = E[Z_{n-1}\mu] = \mu E[Z_{n-1}] = \cdots = \mu^{n} E[Z_{0}] = \mu^{n}.\]

The variance of \(Z_{n}\) is

\[\mbox{Var}(Z_{n}) = \begin{cases} \sigma^2 \mu^{n-1}\left(\frac{1 - \mu^n}{1 - \mu}\right), &\text{if } \mu\neq 1 \\ n\sigma^2, &\text{if } \mu = 1 \end{cases}\]

Remark: the proof for this is more cumbersome, but can be done with conditional variance (see textbook).

\[\mbox{Var}(Z_n) = E[\mbox{Var}(Z_n\mid Z_{n-1})] + \mbox{Var}(E[Z_{n}\mid Z_{n-1}])\]

Generating Function

Definition. The generating function of \(X\) is \(G_X(s) = E[s^{X}]\).

Proposition. Let \(X_{1},\ldots, X_{N}\) be i.i.d. and \(N\) be an integer-valued random variable. Define \(Y = X_1 + \cdots + X_{N}\) and let \(Z_{n}\) be defined as above. Then, the following holds:

• \(G_{X_1 + X_2}(s) = G_{X_1}(s)G_{X_2}(s)\)
• \(G_{Y}(s) = E_N[E[s^{Y} \mid N]] = E_N[(G_{X}(s))^{N}] = G_{N}(G_{X}(s))\)
• \(G_{Z_{n}}(s) = G_{Z_{n-1}}(G(s)) = G_{n}(s)\)

Additional properties:

• \(G_n(0) = P(Z_n = 0)\)
• \(G'(1) = E[X]\)
• \(G''(1) = E[X^2] - E[X]\)
  • \(Var(X) = G''(1) + G'(1) - (G'(1))^2\)