I’m looking for a distribution that is non negative , or has good tail bounds (so non negative with high probability) and maximizes the following property:
$X_1, X_2, …, X_n$ are n IID samples of the distribution
$Y = max(X_1, X_2, …, X_n) $
X = a sample from the distribution
Maximizes $$\frac{E[Y]}{ E[X]}$$
So for instance for a normal distribution, E[Y] is $\theta(\sigma\sqrt{log(n)})$ however for the samples to be non negative with high probability, because of the tail bound on normal distribution, $\mu$ Should grow faster than $\sigma$ so this of $\frac{E[Y]}{ E[X]}$ will be bound by $O(\sqrt{log(n)})$. On the other hand, if I’m not mistaken we can have $\frac{E[Y]}{ E[X]} = \theta(\frac{log(n)}{log(log(n))})$ through binomials, a balls and bins problem setting. However I’m not sure if this is the best that $\frac{E[Y]}{ E[X]}$ can do. I’d appreciate any help.