Suppose $X_{1}, X_{2}, ....$ is a random sample from a probability distribution with mean $\mu < \infty$ and $\sigma^{2} < \infty$. Let $\bar{X}_{n} = \frac{\sum X_{i}}{n}$. From the Central Limit Theorem we can state that: $\sqrt{n}(\bar{X}_{n} - \mu) \xrightarrow {d} N(0,\sigma^{2})$
All right, but now, i want to know how to explain (intuitively) the next statements:
$(\bar{X}_{n} - \mu) \xrightarrow {d} N(0,\frac{\sigma^{2}}{n})$
$\bar{X}_{n} \xrightarrow {d} N(\mu,\frac{\sigma^{2}}{n})$
$ \sum X_{i} \xrightarrow {d} N(n\mu,\sigma^{2})$
Is it because the Slutsky theorem? (about continuous functions)
Thank's a lot!