Let $X_1, X_2, \dots $ i.i.d random variables with the following properties.
(1) $\mathbb{P}(|X_j|>x)= x^{-\alpha}L(x)$, where $\alpha \in (0,1)$ and $L(x)$ is a slowly varying function.
(2) $\lim_{x \rightarrow +\infty} \frac{\mathbb{P}(X_j>x)}{\mathbb{P}(X_j<-x)}\in[0, \infty]$ exists
Let be $$a_n:=\inf\{x: \mathbb{P}(|X_j|>x)<n^{-1}\},$$ $$b_n:=n\mathbb{E}(X_j\mathbb{I}\{|X_j|\leq a_n\})$$
How to prove that $\frac{b_n}{a_n}$ converges to a constant?
Thank you very much in advance!