I have recently been playing around with some change of measure arguments for shifting the mean of a sub-gaussian distribution. It occurred to me however, that sub-gaussianity might not be the natural property to consider, but rather being a member of an exponential family (as this simplifies the likelihood transform induced by an exponential tilting).
Now, I must admit that I do not know much about these, but I encountered a rather miraculous claim on wikipedia, https://en.wikipedia.org/wiki/Natural_exponential_family:
"Every distribution possessing a moment-generating function is a member of a natural exponential family."
As I figured it fair to try to read up a bit before asking stupid questions, I familiarized myself with a paper which seems to prove this result partially:
SAMPSON, Allan R. Characterizing exponential family distributions by moment generating functions. The Annals of Statistics, 1975, 747-753.
Here, the author shows that if the moment generating function is of a particular form, then, the distribution is a member of an exponential family. There however still seems to be a leap, one which I am not able to make by myself, to the claim I quoted; in particular, the hypotheses of the claim seem to be weaker and the conclusion stronger (as we have narrowed down to a natural exponential family)!
I would very much appreciate a reference on this result, preferably with proof (as it seems to me an amazing property which I cannot entirely wrap my head around...) Of course, if anyone knows the proof, I would be even happier if that person answered. Thank you in advance!