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As the classic CLT states Xs follow the same distribution, then the sum of Xs approximate to Normal distribution. But what about several Xs follows different distributions (maybe the same class but not the same value of parameters or totally different kinds)? It seems that the sum will become more central or even more Normal "likely". Maybe some kinds of "Generalized" CLT has interests about that.

Is there any theory doing study about such kind of research? What characteristics if the Xs follow, we will have such results. What kinds of assumptions fails, then we have another results?

We know that sum of the uniforms is triangle distribution, while sum of the Poissons is still Poisson. Sum of stable family (including Normal, Cauchy) is still stable distribution. And about student-t, the sum is not student-t but not knowing what kind it is. But it seem that the fat-tail effect of the sum is thinner.

So could you please tell me some more general rules about such aspects.

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    $\begingroup$ For starters, read about the Lyapunov version and the Lindeberg version of the CLT on the Wikipedia page on the CLT. Then go read some of the references therein for more in-depth knowledge. $\endgroup$ – Dilip Sarwate Nov 1 '13 at 3:01
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    $\begingroup$ You might also do some research on this site, beginning with stats.stackexchange.com/questions/3734/…. $\endgroup$ – whuber Nov 1 '13 at 12:42
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    $\begingroup$ @shusheng Note that you're not summing distributions; you're working out the distribution of the sum of random variables; if they're independent, that's convolution of pdfs/pmfs not sums of cdfs. In terms of the question, one thing of relevance to you might be the Berry-Esseen theorem. $\endgroup$ – Glen_b Nov 17 '16 at 10:00

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