WANTED : Fast-growing 1st order statistics

Question

Is there any distribution that satisfies these two conditions :

1. Bounded variance

2. $E[ $( Maximum of K i.i.d samples )$] = \Omega(\sqrt{K})$?

Or is there any proof that it is not possible?

.

I think if such distribution exists, it should be a very, very heavy-tailed distribution.

As long as I know, the maximum of K standard normal samples grows like $\log K$.

Please help!

Answer 1

I doubt it is possible.

I suspect that $E[ $( Maximum of K i.i.d samples )$] = \Omega(\sqrt{K})$ leads to something broadly like $F(x) = \exp(-c/x^2)$ for $x>0$ (at least in the positive tail); this example has a density of $f(x)=\frac{2c}{x^3}\exp(-c/x^2)$ and a finite mean of $\mathbb E[X]=\sqrt{\pi c}$ but an infinite variance

This example does meet the stated requirement for the expectation of the maximum of $K$ i.i.d. samples, since $Y=\max(X_1,\ldots,X_k)$ has $F(Y) = F(X)^K=\exp(-cK/x^2)$ and mean $\mathbb E[Y]=\sqrt{\pi c K}$ which is $\Omega(\sqrt{K})$