Central Limit Theorem not valid?

Question

According to Central Limit Theorem (CLT), the mean of any i.i.d. sample is Normal distributed (taking $n\rightarrow\infty$ samples).

Let $X_i\sim U(a,b)$.

Then $\bar{X}\sim N$ by CLT.

But as we know, $U$ has a.s. no support outside $(a,b)$ so its mean is never distributed Normal on $\mathbb{R}$? I expect it would be (a,b)truncated-Normal?

Answer 1

You need to look at a more precise statement of the central limit theorem. For i.i.d. random variables $X_i$ with mean $0$ it states that $\frac{X_1 + ... + X_n}{\sqrt{n}}$ (note the division by $\sqrt{n}$ rather than $n$) approaches a normal distribution with variance the variance of $X_i$. In particular, even if each $X_i$ is supported in $(a, b)$, this sum is supported in $(\sqrt{n} a, \sqrt{n} b)$.

Answer 2

Central Limit Theorem says $\sqrt{n}(\bar X-\mu) \stackrel{d}{\rightarrow} N(0,\sigma)$, as $n\rightarrow\infty$. Or you can think of $\bar X$ approaches $N\big(\mu,\frac{\sigma}{\sqrt{n}}\big)$, so the normal distribution shrinks around the mean $\mu$ and the distribution approaches zero rapidly outside any fixed interval around $\mu$. Your statement lost the essential factor $\sqrt n$.