Let $X_1, X_2, X_3, \dots \in \text{Random}(\mathbb{R})$ each be a D3161: Random real number such that

(i)	$X_1, X_2, X_3, \dots$ is an D3358: I.I.D. random collection
(ii)	\begin{equation} \mathbb{E} |X_1|^2 \in (0, \infty) \end{equation}
(iii)	\begin{equation} \mu : = \mathbb{E} X_1 \end{equation}
(iv)	\begin{equation} \sigma^2 : = \text{Var} X_1 \end{equation}

Then \begin{equation} \lim_{N \to \infty} \sum_{n = 1}^N \frac{X_n - \mu}{\sqrt{\sigma^2 N}} \overset{d}{=} \text{Gaussian}(0, 1) \end{equation}

Let $X_1, X_2, X_3, \dots \in \text{Random}(\mathbb{R})$ each be a D3161: Random real number such that

(i)	$X_1, X_2, X_3, \dots$ is an D3358: I.I.D. random collection
(ii)	\begin{equation} \mathbb{E} |X_1|^2 \in (0, \infty) \end{equation}
(iii)	\begin{equation} \mu : = \mathbb{E} X_1 \end{equation}
(iv)	\begin{equation} \sigma^2 : = \text{Var} X_1 \end{equation}

Then \begin{equation} \sum_{n = 1}^N \frac{X_n - \mu}{\sqrt{\sigma^2 N}} \overset{d}{\longrightarrow} \text{Gaussian}(0, 1) \quad \text{ as } \quad N \to \infty \end{equation}