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} |
| (v) | \begin{equation} \overline{X}_N : = \frac{1}{N} \sum_{n = 1}^N X_n \end{equation} |
Then
\begin{equation}
\sqrt{N} \left( \frac{\overline{X}_N - \mu}{\sigma} \right)
\overset{d}{\longrightarrow} \text{Gaussian}(0, 1)
\quad \text{ as } \quad
N \to \infty
\end{equation}