Derivatives of the log moment generating function

Statement

Suppose $M(\theta)=\mathbb E e^{\theta X}$ is finite on an open interval $I$ , and let $g=\log M$ . For $\theta\in I$ ,

g'(\theta)=\mathbb E_\theta X,\qquad g''(\theta)=\operatorname{Var}_\theta(X)\ge0,

where $\mathbb E_\theta$ uses the tilted measure. If $X$ is not almost surely constant, then $g''>0$ throughout $I$ .

Derivation

On compact subintervals of $I$ , nearby exponential moments dominate $|X|^k e^{\theta X}$ for each fixed $k$ . Differentiation under the integral gives $M'=\mathbb E[Xe^{\theta X}]$ and $M''=\mathbb E[X^2e^{\theta X}]$ . Hence $g'=M'/M$ and $g''=M''/M-(M'/M)^2$ . Positivity of the tilted density preserves the property of being almost surely nonconstant, proving strict positivity of the variance in that case.

References

Fithian, Exponential Families (derivatives of the log-partition function).