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Reading "Monte Carlo Statistical Methods" by Robert and Casella, they mention that if $f(x) = h(x) \exp(\langle \theta, x \rangle - A(\theta))$ defines a family of distributions for $X$, parametrized by $\theta$, then $A$ is the cumulant generating function of $h(X)$. It seems like this should be easy to prove if it's true, but I don't see how to proceed. Any ideas/references?

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Integrate with respect to x. The LHS is one. The RHS consists of the product two terms, one being the inverse of the exponential of the cumulant generating function of h, the other being $e^{-A(\theta)}$.

(Also plausible that I am asleep, in which case I apologize)

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