How to show if $X_1$ and $X_2$ both have MGFs (marginally), then the vector $(X_1, X_2)$ has an MGF as well?

Question

Suppose $X_1,X_2$ are jointly distributed r.v.s, not necessarily independent or identically distributed.

I want to show that if $X_1$ and $X_2$ both have MGFs (marginally), then the vector $(X_1, X_2)$ has an MGF.

I am wondering if the final form is:

$$ M_{(X_1, X_2)}(t) = M_{X_1}(t)M_{X_2}(t) $$

Thanks.

Answer 1

Suppose $X_1$ has mgf in interval $|t|< t_1$ and $X_2$ hasd mgf in interval $|t|

Now by Cauchy Schwarz Inequality, $[M_{(X,Y)}(r_1,r_2)]^2=(E(e^{r_1X_1+r_2X_2}))^2\leq E(e^{2r_1X_1})E(e^{2r_2X_2})$. Hence the mgf of $(X_1,X_2)$ exists in $|r_1|