Population Forecasts From the Supercritical Limit

From a Normalized Limit to a Population Forecast

A supercritical branching process has a natural growth scale: if the mean number of children per individual is \(m>1\), generation \(n\) is measured relative to \(m^n\). The normalized population can converge even when the unnormalized population becomes very large. That limit gives a practical way to compare populations and estimate probabilities of crossing thresholds, but it does not make every finite-time forecast an asymptotic one. In particular, the generation at which an observation is made matters.

Work on a probability space \((\Omega,\mathcal{F},\mathbb{P})\). Let \((Z_n)_{n\ge0}\) be a Galton–Watson process with state space \(S=\{0,1,2,\ldots\}\), initial population \(Z_0=1\), and offspring variable \(\xi\). The offspring counts of all individuals are independent copies of \(\xi\). Write \(m=\mathbb{E}\xi\), and let \(\mathcal{F}_n\) contain the population and reproduction information through generation \(n\). Assume \(m>1\). The normalized population is \(W_n=Z_n/m^n\).

For an integer threshold \(K\), the event that the generation-\(n\) population exceeds \(K\) is exactly \(\{W_n>K/m^n\}\), with a strict inequality. If \(K\) is on the scale of \(m^n\), the threshold \(K/m^n\) can approach a positive constant. The limit law of \(W\) can then approximate the exceedance probability. A separate, exact forecast is available for the conditional mean at any observation time.

Proof. Given \(\mathcal{F}_t\), there are \(Z_t\) individuals in generation \(t\). Each has expected \(m^h\) descendants \(h\) generations later, by iterating the offspring mean: the expected number of descendants one generation later is \(m\), and conditional expectation at each subsequent generation multiplies the expectation by \(m\). Linearity of conditional expectation over the \(Z_t\) individuals gives \(Z_t m^h\). If \(Z_t=0\), there are no descendants and both sides are zero. \(\square\)

When the Limit Gives a Growth Ratio

The normalized limit also gives a pathwise statement about growth over a fixed horizon, provided the observation generation itself tends to infinity. On survival, \(W>0\), so the population is eventually nonzero and can be used as a denominator.

Proof. On survival, \(W>0\) almost surely. Since \(W_t=Z_t/m^t\to W\), both \(W_{t+h}\) and \(W_t\) tend to the same positive value \(W\) as \(t\to\infty\), with \(h\) fixed. Thus \(W_{t+h}/W_t\to1\). For all sufficiently large \(t\), \(Z_t>0\), and the definitions give \(Z_{t+h}/Z_t=m^h(W_{t+h}/W_t)\). Taking the limit proves the claim. When \(h=0\), the ratio is exactly \(1=m^0\). \(\square\)

This result justifies \(Z_t m^h\) as a long-run, fixed-horizon growth approximation when \(t\) is large and the process survives. It does not justify deriving that ratio by dividing two limit approximations at a fixed \(t\) while \(h\to\infty\). At fixed \(t\), the denominator \(W_t\) need not be close to \(W\); the normalized limit only controls generations tending to infinity. For an arbitrary observation time, \(Z_t m^h\) remains valid as the conditional mean, but the limit theorem alone does not make it a pathwise approximation to the realized future population.

Scaled Thresholds and Exceedance Probabilities

Proof. On the event \(W\ne x\), almost sure convergence \(W_n\to W\) and ordinary convergence \(x_n\to x\) imply that the two comparisons \(W_n>x_n\) and \(W>x\) eventually agree. Indeed, if \(W>x\), the positive gap \(W-x\) eventually exceeds \(|W_n-W|+|x_n-x|\); if \(W<x\), the corresponding negative gap eventually keeps both comparisons false. The exceptional event \(\{W=x\}\) has probability zero by assumption. Hence the indicators \(\mathbf{1}_{\{W_n>x_n\}}\) converge almost surely to \(\mathbf{1}_{\{W>x\}}\). They are bounded by 1, so the bounded convergence theorem gives convergence of their expectations, which are the stated probabilities. Finally, \(Z_n>K_n\) is equivalent to \(W_n>K_n/m^n\), proving the branching-process version. \(\square\)

At finite \(n\), replacing \(W_n\) by \(W\) is an approximation, not an identity. The proposition explains why it becomes reliable for large generations when the scaled thresholds converge to a continuity point of the limit distribution. If \(W\) has an atom at \(x\), the proposition does not supply the limit at that threshold; extra information about how the finite-generation probabilities approach the atom would be needed.

Fixed Thresholds, Survival, and a Common Error

A fixed absolute threshold behaves differently from a threshold proportional to \(m^n\). If \(K\ge0\) is fixed, then \(K/m^n\to0\), and \(W\) has positive probability at zero whenever extinction is possible. Thus the scaled-threshold proposition cannot be applied at \(x=0\) in that case. The survival statement in the Kesten–Stigum theorem gives the appropriate conclusion directly: on extinction, \(Z_n\) eventually equals zero; on survival, \(W_n\to W>0\), so \(Z_n=m^nW_n\to\infty\). Consequently, for each fixed finite \(K\), \(\mathbb{P}(Z_n>K)\to\mathbb{P}(\text{survival})\). To see the probability limit, the indicators of \(\{Z_n>K\}\) converge almost surely to the survival indicator, and bounded convergence applies.

Common mistake. One might write \(Z_{t+h}\approx Wm^{t+h}\) and \(Z_t\approx Wm^t\), divide, and conclude \(Z_{t+h}\approx Z_tm^h\) for any fixed \(t\). The limit theorem does not guarantee that \(Z_t/m^t\) is close to \(W\) at that fixed time. For fixed \(t\), on survival the limit theorem instead gives \(Z_{t+h}/(Z_tm^h)=W_{t+h}/W_t\to W/W_t\) as \(h\to\infty\), provided \(Z_t>0\). This need not be 1. The correction is to use \(Z_tm^h\) as the exact conditional-mean forecast, or use the pathwise ratio proposition with \(t\to\infty\) and fixed \(h\).

In practice, the mean multiplier answers how the conditional expectation scales; the limit distribution answers how long-run outcomes vary around that scale. Forecasting a threshold requires specifying whether it is fixed or grows with \(m^n\), and whether the desired comparison is strict. A point mass in the limit law can affect a threshold estimate, so continuity at the scaled threshold is a substantive condition, not a formality.