In the one-dimensional case, we have the following fact: (see Existence of the moment generating function and variance for proof)
Proposition: The mgf $m(t)$ is finite in an open interval $(t_n,t_p)$ containing the origin if and only if the tails of $F$ are exponentially bounded, i.e., $P(|X|>x)≤Ce^{−t_0x}$ for some $C>0$ and $t_0>0$.
Is there a similar fact for multivariate random variables? My problem is with proving the "$\leftarrow$" direction as $\{e^{t'X}>e^{t'x}\} \not\subset \{X>x\}$ for $t,X,x \in \mathbb{R}^m, m>1$.
Are there maybe other known sufficient conditions for the mgf to be finite in an neighborhood of zero?
