# Central Limit Theorem with Sample Mean

Let $$X$$ is a random variable from the distribution $$F$$ with mean $$\mu$$.

As we know, $$\bar{X} \equiv \frac{1}{N}\sum \limits_{i=1}^nX_i \xrightarrow{\;\;\;p\;\;\;}\mu$$ by LLN (law of large number).

Here, I am wondering whether there is a kind of central limit theorem ensuring $$\sqrt{N}(\bar{X}-\mu)\xrightarrow{\;\;\;d\;\;\;}N(0, V).$$

Lindeberg–Lévy CLT. Suppose $$\{X_{1},\ldots ,X_{n}\}$$ is a sequence of i.i.d. random variables with $$\mathbb {E} [X_{i}]=\mu$$ and $$\operatorname {Var} [X_{i}]=\sigma ^{2}<\infty$$ Then as $$n$$ approaches infinity, the random variables $${\sqrt {n}}({\bar {X}}_{n}-\mu )$$ converge in distribution to a normal $${\mathcal {N}}(0,\sigma ^{2})$$:
$$\sqrt{n}\left(\bar{X}_n - \mu\right)\ \xrightarrow{d}\ \mathcal{N}\left(0,\sigma^2\right) .$$