Discussion
Suppose the sequence of random variables \(\{X_{n}\}\) converges in distribution to X. Then which of the following is not always true ?
Answer: A
The statement not always true is \(\lim_{n} E[X_{n}] = E[X]\).
Convergence in distribution does not control expectations without uniform integrability. Let \(X_{n} = n\) with probability \(1/n\) and 0 otherwise: \(X_{n} \to 0\) in distribution, but \(E[X_{n}] = 1\) for every \(n\).
The others always hold. (B) is Fatou's lemma in the form valid for convergence in distribution (via Skorokhod's representation). (C) holds because \(|x|/(1 + |x|)\) is bounded and continuous, which is exactly what convergence in distribution handles. (D) holds because a sequence converging in distribution is tight, so \(X_{n}/\sqrt{n} \to 0\) in probability.
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