Behavior of the ratio of max and sum of a collection of iid random variables with power law

Question

I have a collection of random variables ${X_i}, i=1, \ldots, N$.

They're drawn from a Pareto distribution with $\alpha = 2$ and $x_m=1$

$$P(x) = 2x^{-3}$$

My estimate for the max $Z_N$ is $Z_N = \sqrt{N} \Lambda$, where $\Lambda$ is a RV which follows a Fréchet distribution $P(x) = e^{-x^{-2}}\theta(x)$.

The sum is extensive with fluctuations of order $\sqrt{N}$, fluctuations of the max are of the same order of fluctuations of the sum. What does this imply for the behavior of $\frac{Z_N}{S_N}$?

Answer 1

It turns out that its behavior is irregular, since

$$S_N = O(N)$$

$$Z_N = \sqrt{N} \Lambda$$

where $\Lambda$ is a Fréchet random variable with infinite variance. Lots of outliers over a "inverse square root like" behavior for their ratio.