For a probability distribution $$ p(x) \propto x^{-(\mu + 1)} \qquad 0 < \mu < 1$$ both the sum of $N$ i.i.d. samples $S_N$ and the largest element of those samples $x_{\text{max}}$ scale as $N^{1/\mu}$.

It occurred to me that the relative proportion of the largest element $$ \frac{S_N - x_{\text{max}}}{S_N} $$ must therefore (maybe?) for given $\mu$ tend to a constant if $N \to \infty$.

I have ran some numerical simulations to verify this hypothesis, and it seems to hold up, with the relative proportion of $x_{\text{max}}$ tending to some value that gets closer to $1$ as $\mu$ decreases.

I want to know whether this hunch is correct and if so, how the relative proportion depends on $\mu$. Can someone point me to some literature on the subject?

I've thought some more about it, and though I know that $x_{\text{max}}$ follows a Weibull distribution and $S_N$ follows a Levy law (both in the limit $N \to \infty$), it seems that this is not enough to derive the proportion of the largest element, since $S_N$ and $x_{\text{max}}$ are of course correlated.


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