An elementary result from Chebyshev's theorem is that the median and mean of a random variable do not differ by more than one standard deviation. I'm curious if there is a similar result for comparing medians of different distributions:
If $X_1$ and $X_2$ are real random variables with respective medians $\mu_1$ and $\mu_2$, does the covariance matrix $\sigma_{ij}$ impose an upper bound on $|\mu_1-\mu_2|$?
Edit: As noted in the comments, the question as stated definitely has a negative answer. So let me modify it: My initial conception of the problem was that the random variables have some functional relation (in particular, $X_1=X_2^2$). Does this, or some other narrowing of the question, produce a positive claim?