the Kesten-Stigum theorem
/ KES-ten STEE-goom /
For a supercritical Galton-Watson process (mean m > 1) the population, on the event of survival, grows like m^n. The natural normalization W_n = Z_n / m^n is a nonnegative martingale, so it converges almost surely to a limit W. But a martingale can converge to a limit that is degenerate — equal to 0 almost surely — which would mean the normalization is wrong and the true growth is slower than m^n. The Kesten-Stigum theorem says exactly when this does NOT happen: it gives the sharp condition under which W is a genuine, non-trivial random multiple of m^n.
Let sum_k k p_k = m > 1 and let X be the offspring random variable. The Kesten-Stigum theorem states that the following are equivalent: (i) the limit W of W_n = Z_n / m^n satisfies E[W] = 1 (equivalently W_n converges in L^1, equivalently W is not almost surely 0); (ii) the offspring law satisfies the 'x log x' moment condition E[X log+ X] < infinity. Moreover, when these hold, {W = 0} coincides (up to a null set) with the extinction event, so on survival W > 0 strictly. Conversely, if E[X log+ X] = infinity then W = 0 almost surely even though the process may survive — the population still grows, but strictly slower than m^n, and the right normalization is m^n times a slowly-varying correction. The proof goes through a size-biased version of the tree (the spine decomposition): W stays non-degenerate precisely when the offspring along the size-biased spine has the integrability needed for the additive martingale to be uniformly integrable.
Why it matters: this is the definitive theorem on the growth rate of a surviving branching population, and the x log x condition is one of the canonical 'just barely' integrability conditions in probability — slightly more than a finite mean, far less than a finite variance. It generalizes verbatim to multitype branching (with m the Perron-Frobenius eigenvalue), to branching random walk, and to general branching (Crump-Mode-Jagers) processes. The honest content is the failure case: a finite mean m > 1 alone does NOT guarantee that Z_n / m^n has a positive limit. If the offspring tail is heavy enough that E[X log X] = infinity, the naive geometric normalization overshoots and W collapses to zero, a subtle trap when one only checks that the mean is finite.
If offspring has all moments finite (say Poisson or binary splitting), the x log x condition is trivially met and W is a genuine positive limit on survival: Z_n grows like W m^n with W > 0. But construct an offspring law with P(X = k) about C / (k^2 log k) for large k; then m can be finite yet E[X log X] = infinity, and Z_n / m^n -> 0 even though the process survives — its growth is slightly slower than m^n.
W = lim Z_n / m^n is nondegenerate if and only if E[X log+ X] < infinity; otherwise it collapses to zero.
A finite mean m > 1 alone is NOT enough for W to be positive; the extra x log x integrability is essential. When it fails, the population still grows but strictly slower than m^n and W = 0 almost surely.