# n independent, normal, random variable distribution

There is 2 time-series signal and we have to compare the distribution of them. I have heard there is a theory that says for n independent, normal, random variables of a series with many members, the distribution of series must be Gaussian. Could anyone please say the name of the theory or an article I can use to read about it?

• The joint distribution of two or more than two independent normal random variables is multivariate normal, which is called Gaussian, by definition. No need to have many members. Please clarify what you mean by "the distribution of series" and "members". Aug 25 at 8:24

## 1 Answer

Could anyone please say the name of the theory or an article I can use to read about it?

That theorem is the central limit theorem (CLT), and it's definitely probability theory basics textbook material rather than article material (unless you really want to read French or Latin publications by Moivre and Laplace from the 18th and early 19th century; probably you'd find a translation of Lyapunov's 1901 paper, Nouvelle forme du théorème sur la limite de probabilité; I can't read it, because I'm neither good enough at French nor analysis).

But: Sorry, it doesn't work like that:

a theory that says for n independent, normal, random variables of a series with many members, the distribution of series must be Gaussian

Series here means sum.

You have a time-series of samples, that just means a sequence, not a sum. So, the CLT doesn't apply to it, so this is not what you need to look into. You very likely should still learn about the central limit theorem: If you're dealing with random time series, you almost surely will encounter it at some point, and it's one of the very fundamental theorems for multidimensional random variables.

• This answer is incorrect. @AlexTP’s comment hits the nail squarely on the head. Aug 25 at 21:42
• @DilipSarwate Alex and I might have differently interpreted the question, but I don't see how this answer is incorrect. Could you help me pinpoint what's wrong? Aug 26 at 8:37
• The OP states that the $n$ random variables are "independent, normal, (emphasis added by me) random variables of a series" and so they are Gaussian without needing any additional theory to justify the claim. At best, a reference to a textbook that says something like "normal random variables are also known as Gaussian random variables" or vice versa suffices. If you read the last quote from the OP that you have included in your answer literally (distinguishing between sequences and series) then you should cite the result that the sum of independent Gaussian random variables is Gaussian. Aug 26 at 15:16