Asymptotic distribution of the mean?

Question

Question: Given a random sample $X_1 ... X_n$ from a cdf $F$, derive the asymptotic distribution of the the sample mean, $\overline{X}$.

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I am not sure what is being requested here. Does one just need to apply the Central Limit Theorem?

In that case

$$ \sqrt{n}\frac{\overline{X}-\mu}{\sigma}\to_{(d)}\to Z \sim \mathcal{N}(0,1) $$

In that case I rewrite

$$ \sqrt{n}\frac{\overline{X}-\mu}{\sigma} = \frac{\sqrt{n}}{\sigma}\overline{X} + \left(-\frac{\mu\sqrt{n}}{\sigma}\right) $$

So I end up with

$$ \overline{X} \sim \mathcal{N}\left(-\frac{\mu\sqrt{n}}{\sigma} , \frac{n}{\sigma}\right) $$

Does this make sense?

Answer 1

No, the answer should be $\mathcal N(\mu, \sigma^2/n)$. Of course this is assuming the distribution has a finite variance.