Understanding definition of n-dimensional moment generating function

Question

I don't understand, why is the moment generating function not defined for some values? What values could these be?

Also, how big are those intervals $(-c_i,c_i)$ . Are they supposed to be arbitrary small and close to $0$?

Answer 1

Consider the 1-dimensional case. If $X$ has $exp(1)$ distribution then $Ee^{tX}=\int_0^{\infty} e^{-x} e^{tx}dx$ which is $\infty$ for $t \geq 1$. Many standard distributions have infinite MGF outside some interval around $0$.