# Comparing two sum of fractal moments for heavy-tail distribution

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!

• In general, what do you know about $a(n)/b(n)$ if $a(n) \sim O(n^{c_1})$ and $b(n) \sim O(n^{c_2})$ for $c_1,c_2 < 1$? By definition of the big-O notation. I have not tried, but I perhaps you can work it out from the general case. – Therkel Apr 18 '18 at 6:43