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Suppose for a random variable $X$ that $\mathbb{E}X=0$ and $\phi(\lambda):=\mathbb{E}e^{\lambda X}$ exists for all $\lambda\in (0,b)$ for some $b>0.$ Is $\phi$ necessarily non-decreasing in $(0,b)?$ Is it infinitely many times differentiable?

My guesses are NO and YES.

Hedonist
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  • Hint on first question: $X:\Omega\rightarrow\mathbb R$ prescribed by $\omega\rightarrow-1$ is also a random variable. – drhab Aug 10 '15 at 10:05
  • Sorry, I wished to add that $X$ is a centered r.v. Thanks for pointing out a simple counterexample. Question edited to include this. – Hedonist Aug 10 '15 at 11:16
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    Sorry, I just realized that under the condition that $X$ is centered, the mgf must be non-decreasing because $\psi(\lambda):=\log\phi(\lambda)$ is convex and $\psi(0) = \psi^\prime(0)=0$ and log-mgf is a convex function. – Hedonist Aug 10 '15 at 11:20
  • I also was able to show that it is infinitely many times differentiable using the dominated convergence theorem. – Hedonist Aug 11 '15 at 12:54

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