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I'm stuck on a homework problem which requires me that I prove the following:

Say $X$ is a random variable without a finite upper bound (that is, $F_X(x) < 1$ for all $x \in \mathbb{R}$). Let $M_X(s)$ denote the moment-generating function of $X$, so that:

$$M_X(s) = \mathbb{E}[e^{sX}]$$

then how can I show that

$$\lim_{s\rightarrow\infty} \frac{\log(M_X(s))}{s} = \infty$$

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Estimate the expectation $E[e^{sX}]$ from below by $e^{sT}$ times $P(X>T)$. – fedja Nov 9 '12 at 3:46
@fedja Yes! I'll add a solution in a bit. Thanks! – Elements Nov 9 '12 at 3:55
You can write your solution as an answer. – Davide Giraudo Nov 9 '12 at 21:50
up vote 1 down vote accepted

Consider the limit when $s\to+\infty$ of the inequality $$ s^{-1}\log M_X(s)\geqslant x+s^{-1}\log(1-F_X(x)). $$

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