The Normalized Galton-Watson Population Martingale

Why Normalize the Population?

A Galton–Watson process describes successive generations in which each individual independently produces a random number of offspring. If the mean number of offspring is \(m>1\), the expected population grows by a factor of \(m\) per generation. Dividing the actual population by this expected growth scale removes that average increase. The resulting process has a useful conditional-mean property: given the population observed so far, its next value has conditional expectation equal to its current value.

Work on a probability space \((\Omega,\mathcal{F},\mathbb{P})\). Let the state space be \(S=\{0,1,2,\ldots\}\), and let time be the discrete generation index \(n\geq 0\). Fix a finite initial population \(Z_0=z_0\in S\). For each generation \(n\geq 0\) and individual label \(r\geq 1\), let \(\xi_{n,r}\) be that individual’s offspring count. Assume all the \(\xi_{n,r}\) are independent and have the same distribution as a nonnegative integer-valued random variable \(\xi\), with finite mean \(m=\mathbb{E}[\xi]>1\). Define the population recursively by

where a sum with no terms is zero. This is the Galton–Watson process. The filtration \((\mathcal{F}_n)_{n\geq 0}\) records the reproduction observed through generation \(n-1\): \(\mathcal{F}_0\) contains only the initial population, and \(\mathcal{F}_n=\sigma(\xi_{k,r}:0\leq k<n,\ r\geq 1)\) for \(n\geq1\). In particular, \(Z_n\) is \(\mathcal{F}_n\)-measurable, while the offspring counts used to form \(Z_{n+1}\) are independent of \(\mathcal{F}_n\).

The main calculation is the conditional mean of a sum of offspring counts when the number of summands is itself the current population. It is important that the current population is known at time \(n\), but the reproduction counts for generation \(n\) are still independent of the information available at that time.

Proof. For each \(r\geq1\), the event \(\{N\geq r\}\) belongs to \(\mathcal{F}_n\), and \(\mathbf{1}_{\{N\geq r\}}\xi_{n,r}\) is nonnegative. Because \(\xi_{n,r}\) is independent of \(\mathcal{F}_n\), its conditional expectation given \(\mathcal{F}_n\) is \(m\). Conditional monotone convergence therefore gives \(\mathbb{E}[\sum_{r\geq1}\mathbf{1}_{\{N\geq r\}}\xi_{n,r}\mid\mathcal{F}_n]=\sum_{r\geq1}m\mathbf{1}_{\{N\geq r\}}=mN\). The sum on the left is \(\sum_{r=1}^{N}\xi_{n,r}\). It is integrable because its expectation, obtained by taking expectations in the conditional identity, is \(m\mathbb{E}[N]<\infty\). Thus the conditional expectation is well-defined, and the claimed identity holds almost surely. \(\square\)

The Normalized Population Martingale

Proof. We first establish integrability of the populations by induction. The deterministic \(Z_0=z_0\) is integrable. If \(Z_n\) is integrable, apply the lemma with \(N=Z_n\). The recursion gives \(\mathbb{E}[Z_{n+1}\mid\mathcal{F}_n]=mZ_n\), so taking expectations yields \(\mathbb{E}[Z_{n+1}]=m\mathbb{E}[Z_n]<\infty\). Induction now gives \(\mathbb{E}[Z_n]=z_0m^n\), including the case \(z_0=0\).

Each \(Z_n\) is nonnegative and \(\mathcal{F}_n\)-measurable. Since \(m^n>0\), \(W_n=Z_n/m^n\) is also nonnegative and \(\mathcal{F}_n\)-measurable, and \(\mathbb{E}[W_n]=\mathbb{E}[Z_n]/m^n=z_0<\infty\). Finally, using the conditional identity just established,

almost surely for each \(n\geq0\). These are exactly the integrability, adaptedness, and conditional-mean requirements in the definition of a martingale. Nonnegativity follows from \(Z_n\geq0\) and \(m^n>0\). \(\square\)

The identity is conditional: it holds given the entire reproduction history through generation \(n-1\), not just after averaging over all possible histories. Its mechanism is that a known population of \(Z_n\) individuals has expected next-generation size \(mZ_n\). The normalization at generation \(n+1\) divides by one additional factor of \(m\).

What the Martingale Identity Does—and Does Not—Say

The theorem says that the normalized process preserves its conditional mean from one generation to the next. In particular, its unconditional mean remains \(z_0\). It does not say that each realized sequence \(W_0,W_1,\ldots\) is constant, nor does the conditional-mean calculation by itself establish a limit for \(W_n\). When reproduction is random, individual generation sizes can vary substantially; the martingale property concerns conditional expectation, not the direction of each sample path.

The hypothesis \(m>1\) is the setting in which normalization by the mean growth factor is used here. The algebraic conditional-mean identity requires only a finite positive mean \(m\); the stated result assumes \(m>1\). Finiteness matters because the conditional expected offspring count must be finite for the integrability argument. The fixed finite initial population ensures that every \(Z_n\) and \(W_n\) has finite expectation. If \(z_0=0\), then every generation is zero and the theorem still holds, with \(W_n=0\).

Common pitfall: dropping the offspring-mean factor. In the second example, \(Z_0=1\) and \(m=2\). A mistaken calculation might write \(\mathbb{E}[W_1\mid\mathcal{F}_0]=Z_0/2=1/2\), as if the expected next population were \(Z_0\). But each individual has expected offspring count \(2\), so \(\mathbb{E}[Z_1\mid\mathcal{F}_0]=2Z_0=2\). The correct calculation is \(\mathbb{E}[W_1\mid\mathcal{F}_0]=2/2=1=W_0\). In general, the factor \(m\) in the conditional population mean cancels exactly one of the \(m\) factors in \(m^{n+1}\); omitting it breaks the martingale identity.