Discussion
Let \(\{X_{n}, n \ge 1\}\) be a sequence of independent identically distributed random variables with finite (non-zero) fourth moment.
Let \(Y_{n} = (X_{1}^{2} + ......... + X_{n}^{2})/n\)
Which of the following is true ?
As \(n \to \infty\), \(\sqrt{n}(Y_{n} - E[X_{1}^{2}])\).
Answer: C
The limit is a normal random variable.
The \(X_{i}^{2}\) are i.i.d. with mean \(E[X_{1}^{2}]\) and variance \(\tau^{2} = E[X_{1}^{4}] - (E[X_{1}^{2}])^{2}\), which is finite because the fourth moment is. By the central limit theorem,
\[ \begin{aligned} &\sqrt{n}\left(Y_{n} - E[X_{1}^{2}]\right) \\ &\xrightarrow{d} N(0, \tau^{2}) \end{aligned} \]
It is standard normal only if \(\tau^{2} = 1\), which is not assumed, so B is wrong; C covers every \(\tau^{2}\) (including the degenerate normal when \(X_{1}^{2}\) is constant). A chi-square limit arises for \(n(\ldots)^{2}\)-type statistics, not this one, and degeneracy at 0 holds only in that special constant case.
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