Discussion
Let \(\{X_{n}, n \ge 1\}\) be a sequence of random variables such that as \(n \to \infty\), \(X_{n} \to X\) in distribution. Which of the following is not always true as \(n \to \infty\) ?
Answer: B
The statement not always true is \(E[X_{n}^{2} - X^{2}] \to 0\).
Convergence in distribution guarantees \(E\,h(X_{n}) \to E\,h(X)\) for every bounded continuous h. \(\cos x\), \(\frac{x^{2}}{1 + x^{2}}\) and \(e^{i(x + 2)}\) are all bounded and continuous, so options A, C and D always hold.
\(x^{2}\) is unbounded, and the second moments need not converge. For example, let \(X_{n} = n\) with probability \(\frac{1}{n}\) and 0 otherwise. Then \(X_{n} \to 0\) in distribution, but \(E(X_{n}^{2}) = n \to \infty\).
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