For a random variable $X$ let \begin{align} K(t)= \log (E[e^{tX}]) \end{align} This quantity is known as the cumulant generating function.
Question: What is some weak sufficient condition for \begin{align} K^{(2)}(t) \end{align} to be continuous around $t=0$ where $K^{(2)}$ is the second derivative of $K(t)$? Specifically, I am interested in some conditions in terms of moments of $X$.
Some thoughts
Here is one condition that I think is very strong and can be imroved: Suppose that the root-test condition is satisfied \begin{align} \lim_{k \to \infty} \left( \frac{k!}{ |E[ X^k]| }\right)^{1/k}=r>0 \end{align} Then the moment generating function $E[e^{tX}]$ is analytic for $(-r,r)$ and therefore $K^{(2)}(t)$is continuous around $t=0$.
I am curious if it is enough to have $E[X^2]<\infty$ or $E[|X|^3]<\infty$.
Also, let $\phi(t) =E[e^{tX}]$ \begin{align} K^{(2)}(t)=\frac{\phi(t) \phi''(t) - (\phi'(t))^2}{(\phi(t))^2}. \end{align} So this question is equivalent to finding conditions for the continuity of the second derivative of the moment generating function.