: 
independent
(a) If 
are Gaussian, then so is y: in fact,
if y Gaussian exactly, then so must all the 
be Gaussian.
(b) De Moivre in 1733 showed that if 
binomially
distributed, then y was always Gaussian for large x (even though
Gauss didn't exist in 1733).
(c) Laplace, in 1812, showed that de Moivre's result
was generally true for all distributions of 
.
If 
are independent random variables with the same
distribution, then y is asymptotically Gaussian with mean 
and
standard deviation 
(where 
has mean 
and standard
deviation 
, and both 
and 
exist, i.e., are finite).