moment generating functions of positive random variables

Question

If the moment generating function $M_X(\cdot)$ of a random variable is known, is there a way to tell whether $P(X\geq 0) = 1$?

Answer 1

According to Bernstein's theorem, if $g(z)=E\exp(zX)$ is the moment generating function of a nonnegative rv $X$, then the function $x\mapsto g(-x)$ is completely monotone on $[0,\infty)$, and conversely. This means all derivatives of all orders of g$(t)$ must be non-negative.