the martingale limit of the population
In a supercritical Galton-Watson process the size Z_n grows roughly like m^n, but with random fluctuations that never average away. The martingale limit W captures the exact random prefactor: it is the almost-sure limit of the normalized population Z_n / m^n, and it records, in a single random variable, how lucky the early generations were. Two populations with identical offspring laws and the same eventual fate can have very different W, and W remembers that difference forever.
Define W_n = Z_n / m^n where m = E[X] > 1. Conditioning on the n-th generation, E[Z_(n+1) given Z_n] = m Z_n, so E[W_(n+1) given F_n] = W_n: (W_n) is a nonnegative martingale with E[W_n] = 1. By the martingale convergence theorem it converges almost surely to a limit W >= 0. The whole question is whether this convergence is also in L^1, equivalently whether E[W] = 1 rather than W = 0 almost surely; the Kesten-Stigum theorem answers it via the x log x condition. When W is nondegenerate, its law is the unique solution (with mean 1) of the distributional fixed-point equation W = (1/m) sum_(i=1)^X W_i, where the W_i are iid copies of W independent of the offspring count X — a self-similarity inherited from the branching structure. The Laplace transform phi(t) = E[exp(-t W)] satisfies the functional equation phi(m t) = f(phi(t)), tying W back to the generating function f.
Why it matters: W is the canonical example of a martingale limit that can be genuinely random and strictly positive, and it is the prototype for limits arising in self-similar and recursive structures throughout probability — from the height of random trees to the partition functions of branching random walks and directed polymers. The distribution of W is generally not explicit (it has a continuous part on (0, infinity) and an atom of size q at 0 from extinction), and computing its tails is a delicate art (large W corresponds to an unusually fertile root). The standing caveat is the same as Kesten-Stigum: without the x log x condition the 'limit' W is identically zero and conveys no information, so one must verify the integrability before treating W as a meaningful random variable.
For binary splitting with m = 2 and all moments finite, W is a genuine positive random variable on survival: a tree whose early generations branch vigorously locks in a large W, and Z_n is forever about W times 2^n. The atom P(W = 0) equals the extinction probability q, because a lineage that dies has Z_n = 0 = W 2^n from then on.
W = lim Z_n / m^n records the early-generation luck and persists forever; P(W=0) equals the extinction probability.
W is nonnegative and always converges, but it equals zero identically unless E[X log+ X] < infinity (Kesten-Stigum). Its atom at 0 is exactly the extinction probability q; on survival, W is strictly positive only when the integrability holds.