Statistics 612: Superefficiency, Contiguity, LAN, Regularity, Convolution Theorems

(ii) For any sequence of sets {Bn} with Bn ∈ An, whenever Pn(Bn) converges to 0, then EPn (| Xn | 1Bn) converges to 0. The following theorem illustrates the usefulness of uniform integrability; uniform integrability in conjunction with convergence in distribution implies convergence of moments. Theorem 1.1 Suppose that Xn ∈ Lr(P ) with 0 < r < ∞ and Xn →p X. Then, the following are equivalent. (i) {| Xn |r} are uniformly integrable. (ii) Xn →r X; in other words: E(| Xn −X |r) converges to 0. (iii) E | Xn |r converges to E | X |r.