Take two sets of random variables (X1,...,Xd) and (Xn1,...,Xnd ) which depends on (n N ).

We know the moment generating functions of (X1,...,Xd ) and (Xn1,...,Xnd ) exist and that (X1,...,Xd ) converges in distribution to (Xn1,...,Xnd ) if and only if the corresponding moment generating functions converge point-wise in an open neighborhood of (0,...,0) . This means they converge in an open neighborhood of t=0 for each mgf. We can assume this interval includes 1.

Using the above, show that for any constants a1,...,ad R,

(Xn1,...,Xnd)d(X1,...,Xd) if and only if a1Xn1+...+adXndda1X1+...+adXd

Hint: The random variables are not assumed to be independent; instead take (X1,...,Xd) to have a joint distribution, and a joint mgf. The joint mgf of random variables X1,...,Xd is

mX1,...,Xd(a1,...,ad)=E[exp(a1X1+...+adXd)].

which is also the mgf for the random variable a1X1+...+adXd evaluated at 1.