When does the Normal Distribution arise?

#1
Hello guys, Studying the multivariate normal distribution (and the univariate one as well), I found in Muirhead (1982) that the normal distribution arises when "the
observations themselves can be regarded as sums of independent random
vectors or effects
". Mi question is if every time observations regarded as sums of independent random effects follow a (more or less) normal distribution? Mi intuition says no, but I have not been able to find an answer to this. I would be really grateful if someone can provide a counterexample or an explanation to this.

Thanks a lot!

Axs
 

Dason

Ambassador to the humans
#2
No. Just because something is a sum of random variables/effects that doesn't necessary mean that it will automatically be normally distributed.