ThmDex – An index of mathematical definitions, results, and conjectures.
Lindeberg central limit theorem for a standard triangular array
Formulation 0
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}
Proofs
Proof 0
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}
This result is a particular case of R3694: Lindeberg central limit theorem with $M_n = n$. $\square$