Levy's continuity theorem
/ lev-EE /
Levy's continuity theorem is the engine that makes the characteristic-function method a rigorous proof technique for weak convergence, including the central limit theorem. It answers: if the characteristic functions converge pointwise, do the underlying laws converge weakly? Almost — and the theorem says exactly when.
Let X_n have characteristic functions phi_n(t) = E[e^(i t X_n)]. The theorem has two halves. Forward: if X_n converges in distribution to X, then phi_n(t) -> phi(t) for every t, where phi is the cf of X (because e^(itx) is bounded continuous). Converse (the useful half): if phi_n(t) converges pointwise to a limit function g(t) for every t, AND g is continuous at t=0, then there is a probability law mu with cf g and X_n converges in distribution to mu. The continuity-at-0 hypothesis is doing real work: it is equivalent to tightness of the sequence (no mass escapes to +/- infinity), and tightness is exactly what Prokhorov's theorem needs to extract a weakly convergent subsequence; pointwise convergence of cfs then pins the limit down uniquely because cfs separate measures.
This theorem is why the textbook CLT proof works: to show normalized sums S_n converge to N(0,1) you just show their cfs converge to e^(-t^2/2), the standard normal cf, which is continuous at 0 — done. The hypothesis cannot be dropped. If phi_n(t) = e^(-n t^2/2) (the cf of N(0,n)), then phi_n(t) -> 1{t=0}, a limit DISCONTINUOUS at 0; correspondingly the laws N(0,n) spread out and do not converge to any probability law (the mass escapes to infinity). The discontinuity at 0 is the analytic fingerprint of lost tightness.
To prove the CLT for iid X_i with mean 0 and variance 1, write the cf of S_n = (X_1+...+X_n)/sqrt(n) as phi(t/sqrt(n))^n. Using phi(s) = 1 - s^2/2 + o(s^2), we get phi(t/sqrt(n))^n = (1 - t^2/(2n) + o(1/n))^n -> e^(-t^2/2). Since e^(-t^2/2) is continuous at 0 and is the cf of N(0,1), Levy's theorem gives S_n -> N(0,1) in distribution.
The whole CLT reduces to a one-line limit of characteristic functions, justified by the continuity theorem.
Pointwise cf convergence alone is not enough — without continuity of the limit at 0 (equivalently, tightness) mass can leak to infinity and there may be no limiting probability law.