This problem is from Resnick's A Probability Path (chapter 5, Integration and Expectation):
Rapid variation. A distribution tail $1-F(x)$ is called rapidly varying if $$\lim_{t\to\infty} \frac{1-F(tx)}{1-F(t)} = \begin{cases} \infty, & \text{if } 0 < x < 1,\\ 0, & \text{if } x > 1. \end{cases}$$
- Verify that if $F$ is normal of gamma, then the distribution tail is rapidly varying.
- Prove that if $X\geq 0$ is a random variable with distribution tail which is rapidly varying, then $X$ possesses all positive moments, that is, for any $m > 0$, $\mathbb{E}(X^m)$ is finite.
So far I've tried to use another exercise from the same chapter:
(b) For a positive random variable $X$ and for any $\alpha > 0$ we have $$\mathbb{E}(X^\alpha) = \alpha \int_{[0,\infty)} x^{\alpha-1} \mathbb{P}[X > x] \mathrm{d}x.$$ (c) If $X\geq 0$ is a random variable such that for some $\delta > 0$ and $0<\beta<1$ $$\mathbb{P}[X > n\delta] \leq c\cdot\beta^n,$$ then $\mathbb{E}(X^\alpha)$ is finite, for every $\alpha > 0$.
I'm thinking of using part (c) of the problem and, at the same time, dividing the domain $[0,\infty) = [0,1)\cup [1,\infty)$, but I can't see how to use the rapid variation. Any help could be useful. Thank you very much in advance.
For any $a>0$, pick some finite $t_a$ such that, for every $t\geqslant t_a$, $$\frac{1-F(2t)}{1-F(t)}\leqslant2^{-a-1}$$ that is, $$P(X>2t)\leqslant2^{-a-1}P(X>t)$$ Then, for every $n\geqslant0$, $$P(X>2^nt_a)\leqslant2^{-n(a+1)}\,P(X>t_a)\leqslant2^{-n(a+1)}$$ Now, $$E(X^a)\leqslant (t_a)^a+\sum_{n\geqslant0}E(X^a;2^{n+1}t_a\geqslant X>2^nt_a)\leqslant (t_a)^a+\sum_{n\geqslant0}\left(2^{n+1}t_a\right)^a\,P(X>2^nt_a)$$ that is, $$E(X^a)\leqslant (t_a)^a\left(1+2^a\sum_{n\geqslant0}2^{na}2^{-n(a+1)}\right)=(t_a)^a\left(1+2^{a+1}\right)$$ which shows that $E(X^a)$ is finite.