Let and be sequences of Independent Random Variables.
Suppose that and for all .
Let have a third derivative and:
Then
Proof
Write .
By triangle inequality we have
By Taylor’s theorem:
where is between and and similar for
Taking expectations and substracting, the first terms cancel so we are left with:
which has size at most
and the result follows.
Corollary
Let be independent with and with .
Let such that .
Then
where .
Proof
Take normal with mean and .
Then and by the previous theorem we are done.