Assume a heavy tailed distribution whose tail can be approximated as $$P(X\geq x)\sim x^{-\alpha}$$ Consider some fractal moment of iid $X_i$, we have $$\frac{1}{n}\sum_{i=1}^nX_i^{\theta}\sim O(n^{\theta/\alpha})$$ for $\theta<\alpha$ and $n\rightarrow\infty$.
Here I want to consider a more complicated version, compare two different moments as $$\frac{\sum_{i=1}^nX_i^{\theta_1}}{\sum_{i=1}^nX_i^{\theta_2}}$$
Anyone knows how this quantity scales with $n$? Thanks a lot!