Relax assumptions

[raffaelemancuso]

Uncorrelatedness is sufficient for a diagonal variance-covariance matrix, independence is not necessary

[JoramSoch]

Dear @raffaelemancuso, thanks for your PR!

First of all, you are right: Uncorrelatedness is a weaker requirement than independence, such that the Theorem could potentially hold for more cases. However, it is my impression that such theorems are often formulated for independent random variables. Here are a few examples:

• Covariance of independent random variables
• Probability under statistical independence
• Cumulative distribution function of a sum of independent random variables
• Probability mass function of a sum of independent discrete random variables
• Probability density function of a sum of independent continuous random variables
• Moment-generating function of a sum of independent random variables
• Moment-generating function of linear combination of independent random variables

One reason for this might be that independence is more often "known", because independence (but not uncorrelatedness) is assumed by some generative model. Relatedly, uncorrelated random variables are not necessarily independent although the inverse holds true which people often confuse.

I would therefore suggest not to rewrite this Theorem, but to add another Theorem with the relaxed assumptions. What do you think?