he closeness of normal variables with
respect to the linear transformation may be traced to the Central Limit
theorem that we introduce in the present section.
Better versions of CLT are presented in the section
(
Central limit theorem (CLT)
II
).
To see why CLT is true, consider the following calculation.
Suppose the iid variables
have the common density
.
We calculate the Fourier transform (=characteristic
function)
We use independence of
.
We expand the
in powers of
.
We use
,
,...
We apply the transformation
.
We utilize
,
=
,
,
.
For a normal distribution
we
calculate
We use
.
The Fourier transform is an isometry in
.
By comparing (*) with (**) we conclude
It still remains to derive the pointwise convergence. We postpone such issues
until the section (
Vague
convergence
). Advanced versions of CLT are presented in the section
(
Central limit theorem (CLT)
II
).
