for 
,
where each 
is a random sample from a distribution 
with mean
and variance 
, then
This theorem describes the behaviour of a sum of random variables as a random variable:
If for ,
where each is a random sample from a distribution
with mean and variance , then
where 
( ``The mean of the sum is the sum of the means'')
and 
( ``The variance of the sum is the sum of the variances''),
regardless of the shape of 
[as long as 
and 
are finite and well-defined].
Note especially that 
is always
a normal (Gaussian) distribution!
I have used an unconventional notation here
(
instead of 
for the variance)
because K&K use 
for the entropy
and I don't want to create more notational ambiguity
than necessary.