(Convergence lemma
for family of complex numbers) Let
be a family of complex numbers such that

1.
,
,

2.
for a constant
independent of
,

3.
.

Then

Proof

We
calculate
All the potential difficulties within the above calculation are resolved by
observation that the
are small starting from some
according to the condition 1.

Proposition

(Lyapunov CLT) Let
is a family of r.v. such
that
Then

To prove that
we verify the conditions of the proposition
(
Convergence lemma
for family of complex numbers
) pointwise in
for
:
We substitute the Taylor decomposition of
around
in the following
form:
Hence
Since
we have
Then by the proposition (
Lyapunov
inequality
),
Hence,
The
condition
is verified in a similar manner with the use of
.