# What is the General Central Limit Theorem?

General Central Limit Theorem says:

Let $\{(X_{n,j} , 1 ≤ j ≤ n), n ≥ 1\}$ be a triangular array of rowwise independent random variables, set $S_n = \sum_{j=1}^n X_{n,j}, s_n^2 = \sum_{j=1}^n σ_{n,j}^2,n ≥ 1$, where $σ_{n,j}^2 = \mathrm{Var}X_{n,j} , 1 ≤ j ≤ n$, and suppose, without restriction, that

$$\mathrm{E}X_{n,j} = 0 \mbox{ for } \; 1 ≤ j ≤ n, n ≥ 1$$

and that

$$s_n^2 = 1 \;\mbox{ for all } \; n$$.

If every row satisfies the Lindeberg condition follwing

$$\sum_{j=1}^n\mathrm{E}|X_{n,j}|^2\mathbf{1}\{X_{n,j}>\varepsilon \} \rightarrow 0 \mbox{ as } \; n\rightarrow \infty$$

where, for all $\varepsilon > 0$, then

$$\dfrac{S_n}{s_n} →^d N(0, 1)\mbox{ as } \; n\to \infty$$

Am I right about it? Is there anything more I can say about this?

• Where is "For all $\epsilon$ >0" ? – Saty Jul 29 '15 at 4:14
• Yes, thanks for your idea. – Nguyễn Tấn Nhựt Jul 29 '15 at 5:48
• "Is there anything more I can say about this?" Nothing much, except that nobody calls this result "General Central Limit Theorem" but rather CLT for arrays or Lindeberg CLT. – Did Jul 29 '15 at 6:22
• Thank you! I wistfully about CLT for general case. (I am not fluent in English) – Nguyễn Tấn Nhựt Jul 29 '15 at 6:53