Discussion
Let \(\{X_{n}, n \ge 1\}\) be a sequence of independent identically distributed random variables with mean 0 and finite variance \(\sigma^{2} \gt 0\) :
Let \(S_{n} = \dfrac{X_{1} + X_{2} + \cdots + X_{n^{2}}}{n^{2}}\)
Then, which of the following is false ?
Answer: B
The false statement is \(S_{n}e^{S_{n}}\) converges to a normal random variable with mean 0 and variance \(\sigma^{2}\).
\(S_{n}\) is the average of the first \(n^{2}\) terms of an iid sequence with mean 0, so by the strong law of large numbers \(S_{n} \to 0\) almost surely. By continuity, \(S_{n}e^{S_{n}} \to 0\) and \(e^{S_{n}} \to 1\) almost surely, and a.s. convergence implies convergence in probability. That makes A, C and D true.
A degenerate limit 0 is not normal with variance \(\sigma^{2} \gt 0\). (A normal limit would appear only after rescaling, for \(\sqrt{n^{2}}\,S_{n} = nS_{n}\).)
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