Continuity Property for MGFs
If for all and for some , then .
Proof. We will need the following continuity property for moment generating functions.
Suppose are random variables with for and is a random variable with for . Assume for some .
If as for all in , then
The proof is beyond the scope of this course.
Consider . Then
It is enough to prove the theorem for a sequence , i.i.d. with mean and variance . Set . We need to show that
Assume that such that .
Set . By the continuity property for MGFs, it suffices to show that
Note that
We want to show that
We have
Claim.
Proof. Let .
However,
where . So
So
where .
Then we can conclude, because
and hence
Using the property
This transfers convergence of moment generating functions to convergence in distribution. To prove it therefore suffices to establish for every , together with finiteness of the limiting transform for some ; this is how the central limit theorem is proved.
Related
Stated in
- Theorem 5.9 (Continuity Property for MGFs)§5.2 Central Limit Theorem