The cumulant generating function $K(t)$ of a random variable is defined as $$K(t) = \log \mathbb{E} [e^{Xt}]$$ for any $t$ such that the exponential moment is finite.
In Wikipedia, it is said that:
The cumulant generating function K(t), if it exists, is infinitely differentiable and convex, and passes through the origin. Its first derivative ranges monotonically in the open interval from the infimum to the supremum of the support of the probability distribution, and its second derivative is strictly positive everywhere it is defined, except for the degenerate distribution of a single point mass.
There is not reference on Wikipedia where this is discussed and proved and I have been unable to find a satisfactory one, even if those facts seems to be common knowledge. Are there any textbook or paper where the proofs are detailed?