If $X$ is a random variable, then it's moment generating function is the function
$$ t \rightarrow E(\exp(tX)) $$
which takes values in $]0,+\infty[$.
For a sequence of random variables $X_n$ point convergence of their MGFs to the MGF of a limit distribution intuitively seems like an extremely strong convergence.
In particular, it should be stronger than the weak convergence: so that the following statement is true:
- If there exists $r$ such that: $\forall t \in [-r,r]$ then $E(\exp(tX_n)) \rightarrow E(\exp(tX))$, then $X_n \rightarrow X$ weakly
I have found three references that say that this is true, but don't offer detailed proofs (Kallenberg, second edition theorem 5.22, http://www.math.uni-bremen.de/~osius/download/papers/MAP33Osius.pdf lemma 2, http://www.math.uah.edu/stat/expect/Generating.html). Can anybody give a reference for a full detailed proof ?
Edit: I have finally found a proof in Billingsley. see answer below