Asymptotic joint distribution of the sample medians of a collection and a sub-collection of i.i.d. random variables

Question

Let $X_1,\dots, X_n$ denote i.i.d. random variables, with a smooth density $f$ having unique median $m$.

It is known that the sample median $M_n$ of $X_1,\dots, X_n$, $M_n = \operatorname{med}(X_1, \dots, X_n)$, is asymptotically normal with mean $m$ and variance $\frac 1 {4 n f(m)^2}$. (Or phrased more precisely, $Z_n = (M_n-m)\sqrt{4 n f(m)^2}$ approaches a standard normal distribution; e.g. Grimmet & Welsh, Probability, an introduction, problem 8.6.16).

Now, for $k \leq n$, let $M_k$ denote the sample median of the first $k$ variables, $M_k := \operatorname{med}(X_1, \dots, X_k)$.

What is known about the asymptotic joint distribution of $(M_k, M_n)$ as $k \rightarrow \infty$ and $n \rightarrow \infty$ (perhaps for $k = \lfloor \alpha n \rfloor $ for some $0 < \alpha < 1$)?

I am able to obtain a rather involved non-asymptotic expression for the joint distribution of $(M_k, M_n)$ but before trying to derive an asymptotic expression, I wonder what is known about this.

Update It is natural to expect that $M_k$ and $M_n$ are correlated Gaussian random variables with correlation coefficient $\sqrt{\alpha} \approx \sqrt{k/n}$. Is it true?

Answer 1

If each $X$ is Bernoulli with probability $p$, then the correlation between $M_k$ and $M_n$ has the following graph for $k=21$, $n=41$:

So also for smooth distributions approximating the Bernoulli distributions, the correlations of medians would depend on $p$, and not just on $k$ and $n$. In general, the correlations of medians would depend on the parent distribution and not just on the sample and subsample sizes.