Imagine you sample $n$ number with replacement uniformly from the integers $1,\dots, n$. Let $X$ be the minimum of these samples. I am interested in $\mathbb{E}(X)$ but with a twist. All I know is that the samples are uniform and pairwise independent.
What bounds can one give for $\mathbb{E}(X)$?
If we generalize this to $k$-wise independence, for $k \geq 2$, what can we say? We can assume $n$ is large.
When you only assume pairwise independence, it is possible that two variables will completely determine a 3rd variable. So imagine this is the case and the 3rd variable is basically chosen so that the overall sample of size 3 looks as "uniform" as possible, i.e. the 3rd variable is negatively correlated with the other $2$. Clearly this will lower ${\mathbb E}X$. And, at least when $n$ is small (like $n = 3$) then ${\mathbb E}X$ can be reduced by a fairly large percentage.
My intuition is that as $n$ increases, you can split into subsets of variables of size $3$ and do this same negative correlation trick, and ${\mathbb E}X$ will still be lowered by a non-negligible percentage in the limit as $n \to \infty$.
