I have a somewhat advanced question involving the role of slowly varying functions and their relation to moments. I want to use them to derive certain results for domains of attraction.
My problem is the following.
Let $L(x) = E[Y^2; |Y|\le x] $ and assume that L(x) is slowly varying to $\infty. $ This means that for any $t > 0, $ we have
$$ \lim_{x\to \infty} \frac{L(tx)}{L(x)} = 1. $$
Fix $p\in [0, 2). $ I can prove (it is some work, but not a major task) that $$ \lim_{x\to\infty} \frac{x^{2-p}E[|Y|^p; |Y|>x]}{L(x)} = 0 \hskip 10pt (*) $$
I would like to prove that $E[|Y|^p] < \infty $ for $p \in [0, 2). $
I am aware that there are certain theorems (Karamata type theorems) that use arguments related to the survival function $G(x) = Pr[Y> x] = 1-F(x) $ varying slowly to infinity to derive the existence of certain moments. But, I wonder if what I know, i.e., (*), is enough.
My line of attack is the traditional one, i.e., $$E[|Y|^p] = E[|Y|^p; |Y|\le x] + E[|Y|^p; |Y|>x] $$ Now the first term on the right-hand side is clearly bounded and I was hoping to use what I learned above to prove that also the second term on the right-hand side is bounded, but I seem to run in some problems. For instance, I could use (*), and write $$ E[|Y|^p; |Y|>x] = x^{2-p}E[|Y|^p; |Y|>x]/L(x) \cdot \frac{L(x)}{x^{2-p}}. $$ The first term on the right goes to 0 as $x \to \infty, $ however, I have an $\frac{\infty}{\infty} $ issue on the second term. Am I approaching the issue in the wrong way or am I missing something else?
As usual thank you to whoever could chip in some words of wisdom.
Maurice