Moment generating function is finite given limsup property

Question

For a random variable $X$ define the moment-generating function $M_X(t) = \mathbb{E}[e^{tX}]$. If it is known that

$$\limsup_{x\to\infty}\frac{\log\mathbb{P}(X>x)}{x} =- c < 0,$$

how can it be shown that $M_X(t) < \infty$ for all $t \in [0, c)$? (source, Ex. 2)

Answer 1

$Ee^{tX} =\int_0^{\infty} P\{e^{tX }>x\}\, dx$. It is enough to show that $\int_M^{\infty} P\{e^{tX }>x\}\, dx <\infty$ for some $M$. Let $t<c$ and $\epsilon <c-t$. By hypothesis there exists $x_0$ such that $\log P\{X>x\} <x( -c+\epsilon)$ for $x > x_0$. Hence $P\{X>x\} <e^{x(-c+\epsilon)}$ and $\int_M^{\infty} P\{e^{tX }>x\}\, dx =\int_M^{\infty} P\{X >(\log \, x) /t\}dx<\int_M^{\infty} x^{\frac {-c+\epsilon} t} dx<\infty$ because $(-c+\epsilon) /t <-1$.