I'm a researcher in social science and I have encountered the following math formulation of a problem in my field. Note that I have also posted on math.stackoverflow, but given that this seems to be an open-ended problem, I'm posting here also. Please let me know if I should delete the double post from either site (otherwise I will definitely update both sites when I receive an answer). Thank you!
Let $x_1,x_2,...,x_n,x_{n+1}$ be $n+1$ i.i.d. random variable with non-negative support. Let $$z_k\equiv \min\{x_1,...,x_k\}.$$
In simulations, I find $Cov(z_{n+1},z_n)$ to be very close to $Var(z_{n+1})$ and very different from $Var(z_{n})$, as long as $n>>1$. I have tried this for many distributions (with non-negative support): $Cov(z_{n+1},z_n) / Var(z_{n+1}) \approx 1$ and the approximation gets better as $n$ increases, even for small $n$ such as $n=5$. On the other hand, $Cov(z_{n+1},z_n)/Var(z_n)$ is very far from 1.
How can I formalize this? That is, I'm looking for some kind of bounds on how the approximation improves with $n$.
