Discussion
Let \(\{X_{n}\}\) be a sequence of random variables such that :
\(P(X_{n} = -n^{1/2}) = P(X_{n} = n^{1/2}) = 1/2n\) and \(P(X_{n} = 0) = (n - 1)/n\).
Let \(S_{n} = \sum_{i=1}^{n} X_{i}\).
Then, which of the following statements is not true ?
Answer: C
The statement that is not true is \(S_{n}/\sqrt{n} \to N(0, 1)\), taking the \(X_{n}\) independent as the question intends.
Each \(X_{n}\) has \(E(X_{n}) = 0\) and \(V(X_{n}) = n \cdot \tfrac{1}{2n} \cdot 2 = 1\), so A and B hold, and \(V(S_{n}/\sqrt{n}) = 1\). But the Lindeberg condition fails: almost all the variance of \(X_{k}\) sits on the rare values \(\pm\sqrt{k}\), which are of the same order as \(\sqrt{n}\) for \(k\) near \(n\).
The characteristic function of \(S_{n}/\sqrt{n}\) has logarithm close to \(-\sum_{k \le n}\tfrac{1}{k}(1 - \cos(t\sqrt{k/n}))\), which tends to \(-\int_{0}^{1}\tfrac{1 - \cos(t\sqrt{u})}{u}du\). That is a genuine limit but not \(-t^{2}/2\), so the limit Z exists and is not Gaussian (D), and C is false.
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