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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 $$\sqrt{N}(\bar{X}-\mu)\xrightarrow{\;\;\;d\;\;\;}N(0, V).$$

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1 Answer 1

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Sure there is

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 ${\mathcal {N}}(0,\sigma ^{2})$:

$$\sqrt{n}\left(\bar{X}_n - \mu\right)\ \xrightarrow{d}\ \mathcal{N}\left(0,\sigma^2\right) .$$

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