Yaglom's theorem
/ YAH-glom /
If a branching population is destined to die out — as it is in the critical and subcritical cases — a natural question survives even after the process does not: GIVEN that the lineage is still alive at generation n, how big is it? Yaglom's theorem answers this by describing the limiting law of the population size conditioned on non-extinction. It reveals a clean universal shape hiding inside processes that, unconditionally, just vanish.
Critical case (m = 1, finite offspring variance sigma^2 > 0): conditioned on survival, the population is of order n. Precisely, the law of Z_n / n given {Z_n > 0} converges to an exponential distribution with mean sigma^2 / 2; equivalently Z_n / n converges in distribution to Exponential(2 / sigma^2). This pairs with the survival asymptotics P(Z_n > 0) about 2 / (sigma^2 n): a critical lineage survives to generation n with probability of order 1/n, and if it does, it carries of order n individuals. Subcritical case (m < 1): conditioned on survival, the population size does NOT grow — Z_n given {Z_n > 0} converges in distribution to a proper (non-degenerate, finite) random variable on the positive integers, the Yaglom quasi-stationary law, with a generating function determined by the offspring law; survival probability decays geometrically like c m^n. The proofs analyze the iterates f_n of the generating function near the fixed point s = 1, where the convexity of f controls the rate of approach.
Why it matters: Yaglom's theorem is the foundation of quasi-stationarity — the idea that a doomed population, viewed before its inevitable death, settles into a stable conditional profile. This underlies models of endangered populations, metastable chemical and epidemic systems, and the analysis of critical trees (the conditioned critical tree, of size of order n, scales to Aldous's continuum random tree). The honest subtleties: the exponential limit in the critical case requires finite offspring variance (heavy-tailed critical offspring give a stable-law limit with a different scaling), and the conditioning {Z_n > 0} is essential — without it the limit is the point mass at 0. Conditioning on a probability-zero-in-the-limit event is exactly what extracts the nontrivial structure.
Critical binary splitting (0 or 2 children with probability 1/2 each) has sigma^2 = 1. Yaglom's theorem says that given the family is still alive after n generations, its size divided by n is approximately exponential with mean 1/2. So a rare survivor at generation 100 typically numbers around 50, not 1 — survival forces unusual size.
Conditioned on surviving to generation n, the critical population size divided by n is approximately exponential.
The exponential limit holds for critical processes with FINITE offspring variance; with infinite variance the conditioned limit is a stable law on a different scale. The conditioning {Z_n > 0} cannot be dropped — without it the limit is just the point mass at 0.