Suppose $(X_n)_n$ is a sequence of r.v's with p.d.f's $|x|^{-3}$ outside of $(-1,1)$. I am trying to show that $Y:=\frac{\sum_i^n X_i}{\sqrt{2n\log n}}$ standard normal r.v.
Solution: It is easy to see that $E[X_i]=0$ (using properties of odd function integrals). Similarly we get the variance to be $Var(X_i)=\int_Rx^2/|x|^3=\lim_{x\to\infty}\ln(x)$. Now by CLT, we have that $Y$ is normal with mean $\frac{nE[X_i]}{\sqrt{n\log n}}=0$ and variance $\frac{Var(X)}{\log(n)}$. When you work this out you get the variance to be 1, but this only as a result of an argument that has an "$\infty/\infty$". I don't really like this argument, (although it is correct). I was wondering if I could see a solution that utilizes characteristic functions and the continuity theorem. This method is likely to avoid indeterminate forms.