Under what conditions are the cumulants of the sufficient statistic finite for an exponential family?
If we have $$ p(x \mid \theta) = \exp(\theta \cdot T(x) - A(\theta)) $$ then the derivatives of $A(\theta)$ wrt $\theta$ give the cumulants of $T(x)$. If $A(\theta)$ is finite, then are all of the cumulants of $T(x)$ finite? Ideally, I'm looking for a reference or proof.
I am trying to establish the conditions under which $A(\theta)$ can be approximated by its Taylor polynomial.