Distribution's shrinking rate for the mean of infinite variance variables

Question

Consider a set of $i.i.d.$ (positive) random variables $\{X_i\}_{i=1}^N$. Each variable $X_i$ has a distribution with finite mean but infinite variance, so $P_{X_i}(x) \sim \frac{1}{x^{\alpha +1}}$ with $1< \alpha <2$.
If we consider the variable $W_N = \frac{(\sum_{i=1}^N (X_i - \langle X\rangle))}{N}$, the variance of $W_N$ goes to 0 for $N \rightarrow \infty$ (for the law of large number)
I want to get the scaling exponent which determine the leading term of the variance's decrease rate of $W_N$.
In other words, for $N \rightarrow \infty$ $\int_{-\infty}^{\infty} w^2P_{W_N}(w) dw = aN^{-b} + o(N^{-b})$, with $a>0, b>0$. I want to get the value of $b$ (that will depend on $\alpha$ value) exponent (and possibly also of $a$).
Any idea?