Central Limit Theorem Variant

Question

Given an input of 4096-dimensional independent and identically distributed (iid) vectors, and an output obtained by performing weighted summation on these iid vectors using coefficients that are also iid, can it be proven that the resulting output conforms to a normal distribution?

Answer 1

No, it cannot be proven- in fact, it is false in general. For one, this is a problem of finite numbers of random variables, and the Central Limit Theorem deals with the behavior of random variables in the limits of sample sizes. Even for something as simple as taking an average of a set of iid random variables, where the CLT applies in the limit, the distribution of the mean will not be normal for any finite sample size unless the underlying distribution is itself normal.

If you're asking whether the CLT would apply to, say, the average of values generated from that process as the sample size grows without bound, the answer is still no in general. The CLT requires that the random variables have finite first and second moments. You would need to make the additional assumption that the product of the two distributions has finite first and second moments.