I have a question about a statistical concept and would like to gather your thoughts on it. As we know, the Central Limit Theorem states that under certain conditions, the sum or average of independent and identically distributed random variables tends to a normal distribution as the sample size increases. However, I am pondering the reverse of this theorem:
If the sum of random variables $x_1, x_2 ... x_n$ exhibits a normal distribution, can we infer that these variables are independent? In other words, is a sum that conforms to a normal distribution sufficient to indicate the independence of the variables?
Do we have any idea or reference?
The answer is NO. It is not possible to grant that if $S = \sum_{i=1}^{n} x_i$ has normal distribution, then $x_1, \ldots, x_n$ are independents.
In fact, if you check the Wiki page of the CLT, you will find that it exist several versions of the CLT and the "CLT under weak dependence" does not require the independence of the samples.