Weak convergence and infinite square mean

Question

$(\Omega , \mathscr{F}, P)$ probability space, $X_n$ are i.i.d., $S_n:=X_1+\cdots X_n$

Suppose $a_n, b_n>0 $ with $\frac{S_n-a_n}{b_n}$ weakly convergent to $N$ standard normal distribution and $E[X_1^2]=∞$. Then for all $x>0$, $P(|X_1|>b_nx)\to 0$

I think $b_n\to \infty$, but I can't prove.

Answer 1

Assuming that $E|X_1|<\infty $, $b_n \to \infty $. See Convergence of Types Theorem In Probability by Leo Breiman. (Use the hypothesis in conjunction with SLLN's)