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Let $X_1,X_2,\ldots,X_n$ be i.i.d. RVs with $E(X_{i})=\mu$ and $V(X_{i})=\sigma^2$, $\sigma <\infty$.Is it possible to find real sequences $a_{n}$ and $b_{n}$ such that $a_{n}(\bar{X}^3_{n}-b_{n})$ converges in distribution to a non-degenerate RV?Here, $\bar{X}_{n}=\frac{1}{n}\sum_{i=1}^{n} X_{i}$.

Ans: According to the suggestion

$\sqrt n\bar{X}_{n} \xrightarrow {a} Z \sim N(\mu,\sigma^2)$ as $n \rightarrow \infty$.

Now we are applying continuous mapping theorem for $g(\bar{X}_{n})=\bar{X}^3_{n}$ and get,

$(\sqrt n\bar{X}_{n})^3 \xrightarrow {a} Z^3 \sim N^3(\mu,\sigma^2)$. Hence,

$a_{n}=(\sqrt n)^3$

$b_{n}=0$.

But, if we want to find some variable which follows standard normal distribution then please give me some hint about how to proceed?

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  • $\begingroup$ (1) What is "$\bar X_n$" and how might it be related to the $X_i$? (2) How do you even know $\bar X_n$ has a third moment or that $\bar X_n^3$ has a variance? $\endgroup$
    – whuber
    Aug 21, 2018 at 20:20

1 Answer 1

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Hint: $$ \overline{X_n}^3 =n^{-3/2}\left(\sqrt n\overline{X_n}\right)^3. $$ Then combine the central limit theorem with the continuous mapping theorem.

Notice that no higher moments are required.

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  • $\begingroup$ Could you please explain why the CLT applies? What happens when, say, the $X_i$ are iid with a Student t distribution with 2 df? $\endgroup$
    – whuber
    Aug 22, 2018 at 11:23
  • $\begingroup$ @whuber The central limit theorem gives the convergence of $\sqrt n\overline{X_n}$ to a central normal law with variance $\sigma^2$, hence $n^{3/2}\overline{X_n}$ converges in distribution to $N^3$, where $N$ has a central normal law with variance $\sigma^2$. $\endgroup$ Aug 23, 2018 at 11:57

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