Given a cumulative distribution function (cdf) $F:\mathbb{R}\rightarrow[0,1]$, its moment generating function (mgf) is defined by $$M_F(t)=\int e^{tx} \, dF(x),\ \forall\ t\in\mathbb{R}.$$ Wiki (https://en.wikipedia.org/wiki/Moment-generating_function) mentions three important properties of $M_F$:
- $M_F(t)\geq 0$ for all $t$
- $M_F(0)=1$
- $\log(M_F(t))$ is a convex function in $t$
My question is, if these three properties also define a moment generating function? That is, if we take an arbitrary function $M:\mathbb{R}\rightarrow\mathbb{R}$ that satisfies those three properties, does there exist a cdf $F$ such that $M=M_F$? If not, what else do we need on $M$ for it to be a mgf of some cdf?
I would also like to ask similar questions for multivariate mgf, as well as characteristic functions. Any reference would be helpful, thanks.
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Edit:
I suddenly noticed a characterization result for characteristic function, which is called Bochner's theorem (https://en.wikipedia.org/wiki/Bochner%27s_theorem).
Maybe a silly question, does Bochner's theorem also characterize (multivariate) moment generating functions?