Let $\{ X_{n, m} \}_{n \geq 1, \, 1 \leq m \leq n}$ be a D5164: Random real standard triangular array such that
(i) | $X_{n, 1}, \dots, X_{n, n}$ is an D2713: Independent random collection for each $n \geq 1$ |
(ii) | $\mu_{n, m} : = \mathbb{E} X_{n, m} \in \mathbb{R}$ |
(iii) | $\sigma^2_{n, m} : = \text{Var} X_{n, m} \in (0, \infty)$ |
(iv) | \begin{equation} s^2_n : = \sum_{m = 1}^n \sigma^2_{n, m} \end{equation} |
(v) | \begin{equation} \forall \, \varepsilon > 0 : \lim_{n \to \infty} \sum_{m = 1}^n \mathbb{E} \left( \left| \frac{X_{n, m} - \mu_{n, m}}{s_n} \right|^2 I_{\left\{ \left| \frac{X_{n, m} - \mu_{n, m}}{s_n} \right| > \varepsilon \right\}} \right) = 0 \end{equation} |
Then
\begin{equation}
\sum_{m = 1}^n \frac{X_{n, m} - \mu_{n, m}}{s_n}
\overset{d}{\longrightarrow} \text{Gaussian}(0, 1)
\quad \text{ as } \quad
n \to \infty
\end{equation}