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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?

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  • $\begingroup$ Where is "For all $\epsilon$ >0" ? $\endgroup$ – Saty Jul 29 '15 at 4:14
  • $\begingroup$ Yes, thanks for your idea. $\endgroup$ – Nguyễn Tấn Nhựt Jul 29 '15 at 5:48
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    $\begingroup$ "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. $\endgroup$ – Did Jul 29 '15 at 6:22
  • $\begingroup$ Thank you! I wistfully about CLT for general case. (I am not fluent in English) $\endgroup$ – Nguyễn Tấn Nhựt Jul 29 '15 at 6:53

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