uniform integrability and convergence of moments
You already met uniform integrability in Volume I as the condition that makes an L^1-bounded martingale converge in L^1 and that rescues the dominated convergence theorem without a dominating function. Here we revisit it for one specific job: it is the exact bridge from convergence in distribution to convergence of moments. Knowing X_n => X tells you nothing about whether E[X_n] -> E[X]; uniform integrability is precisely the missing hypothesis that makes that implication true.
A family {X_i} of random variables is uniformly integrable if sup_i E[ |X_i| ; |X_i| > K ] -> 0 as K -> infinity, that is, the tail contribution to the mean is small uniformly across the family. Equivalently (de la Vallee-Poussin), there is a function phi with phi(t)/t -> infinity such that sup_i E[phi(|X_i|)] < infinity, for example a uniform bound on E[|X_i|^{1 + delta}] for some delta > 0 implies uniform integrability. The key theorem: if X_n => X (convergence in distribution) AND the family {X_n} is uniformly integrable, then E[X_n] -> E[X], and X has a finite mean. More generally, if {|X_n|^p} is uniformly integrable and X_n => X, then E[X_n^p] -> E[X^p]; uniform integrability of the p-th powers is exactly what upgrades convergence in law to convergence of the p-th moment. This is also the statement that Wasserstein-p convergence equals weak convergence plus uniform integrability of the p-th powers.
Without uniform integrability the implication simply fails, and the failure is the canonical cautionary example of the whole field: X_n = n with probability 1/n and 0 otherwise has X_n => 0 yet E[X_n] = 1 forever, because the family is not uniformly integrable (the runaway mass at n carries a fixed amount of mean). So whenever someone passes from a distributional limit to a statement about expectations, variances, or any moment, look for the uniform-integrability (or a uniform higher-moment) hypothesis: it is doing all the work, and dropping it is a genuine error, not a harmless simplification.
Let X_n be N(0, 1 + 1/n), so X_n => N(0,1) in distribution. The family {X_n^2} is uniformly integrable because sup_n E[X_n^4] = sup_n 3(1 + 1/n)^2 <= 12 < infinity (a uniform fourth-moment bound). Hence Var(X_n) = E[X_n^2] -> E[X^2] = 1: weak convergence plus UI of the squares delivers convergence of the variance.
A uniform fourth-moment bound makes the squares uniformly integrable, so the variances converge along with the distributions.
Convergence in distribution alone never gives convergence of any moment. The X_n = n w.p. 1/n example has X_n => 0 but E[X_n] = 1 always. Uniform integrability of the relevant power is the non-negotiable extra hypothesis.