Discussion
Let \(\{X_{n}, n \ge 1\}\) be a sequence of independent identically distributed random variables with mean '0' and variance '2'. Let :
\[ S_{n} = \sum_{i=}^{n} X_{i}^{2}. \]
Which of the following statements is true ?
Answer: B
Sn/n converges to 2 almost surely.
The variables \(X_{i}^{2}\) are i.i.d. with mean
\[ E(X_{i}^{2}) = Var(X_{i}) + [E(X_{i})]^{2} = 2 + 0 = 2 \]
By the strong law of large numbers, \(\frac{1}{n}\sum_{i=1}^{n} X_{i}^{2} \to 2\) almost surely. (The paper prints the lower limit as 'i =', meaning i = 1.)
So the limit is 2, not 0. \(S_{n}/\sqrt{n}\) behaves like \(2\sqrt{n}\) and tends to infinity, not 0, and \(\sqrt{n}S_{n}/2\) also diverges, so it cannot converge to a standard normal.
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