Existence of pdf and mgf

Hello,

I have some struggles understanding existence implications between the moment generating function and a probability distribution.

Going from a pdf / cdf to an MGF is OK, the MGF M(t)exists if the expectation E(e^{tx}) exist at some neighborhood of 0.

The other way, however. Given a function claiming to be an MGF, for example M(t) = t or M(t)=t/(1-t), for |t|<1. How can one know if it exists a corresponding pdf?

Obviously, if one calculate M''(t) and evaluate it at 0 and finds it negative, then M(t) can not be an MGF for a probability distrubution. What I am looking for is a general rule to check this.

Also, say that one can decide that an MGF is valid, i.e., there is a corresponding pdf. Then the pdf can be found by inverse Laplace transforming M(-t). Is this in convenient also for distribution that can take negative values? Or does it exist any clever tricks for finding the pdf?

Thanks,

Olw
.

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