Is there any truth to the statement "the median of medians converges to the true median?".
While it's false that the median of medians is the median, is there a way to make this true asymptotically? I'm happy to take on regularity conditions on the underlying distribution (like continuous density function, etc).
More precisely, this is my setup. I have iid random samples $x_{i,j}$ of some random variable $X$.
For each $i$, I can take the median over $j$.
$$ m_i(n) := \text{median}(x_{i,j} | j \leq n) $$
And then take the median over those
$$ m(k,n) := \text{median}(m_i(n)| i \leq k) $$
Is it true that
$$ \lim_{k,n \to \infty} m(k,n) = \text{the actual median of } X $$
under some conditions on $X$?
And same question where median is replaced by other quantiles of $X$.
(though a counterexample with a non-super-weird $X$ (say continuous with finite mean and variance) would make me even happier)
I found this reference from another question, https://www.math.ucla.edu/~tom/papers/unpublished/meanmed.pdf
where it claims that the distribution of sample quantiles is asymptotically normal. So I guess the correct statement is that "average of quantile converges to quantile". IIUC, a counterexample to "quantile of quantile converges to quantile" would be greatly appreciated then!