I am stuck on the following problem:
Let $\{X_n\}$ be a sequence of iid random variable with density: $$f(t)=c \frac{1}{|t|^{3+\alpha}}1_{\{|t| \ge 1 \}}\quad; t \in \mathbb{R}, \quad \alpha > 0$$
Choose $\{B_n\}$ to be a sequence of positive reals such that $$\{B_n\} \to \infty \quad \text{as} \quad n \to \infty$$
Provide the necessary and sufficient conditions on the sequence $\{B_n\}$ so that $$\lim_{n \to \infty} \frac{X_1+...+X_n}{B_n}=N(0,1)$$ in distribution.
I am not sure how to approach this problem. Should I be trying to calculate the characteristic function for $\displaystyle \frac{X_1+...+X_n}{B_n}$ and then attempting to deduce conditions based on that?
This is a classical situation in the sense that we have iid random variables with finite second moments (see here). We know that $$ \frac{X_1+\ldots+X_n}{\sqrt n}\xrightarrow{d}N(0,\sigma^2) $$ as $n\to\infty$, where $\sigma^2=\operatorname{Var}X_1$.
So we need to make sure that the variance of $$ \frac{X_1+\ldots+X_n}{B_n} $$ converges to $1$ as $n\to\infty$. The norming sequence $\{B_n\}$ is going to be equal to $\sigma\sqrt n$ for each $n\ge1$ with some positive constant $\sigma$ that we need to find. This constant $\sigma$ is going to depend on $\alpha$.
I hope this helps.