Here is my problem : $Z_i$ a sequence of i.i.d. random variables on the same probability space with median k.
$M_n = |{i = 1 . . . n : Z_i ≤ k}|$ and $P_n = |{i = 1 . . . n : Z_i > k}|$.
Can we describe $L_n$ as a sum of i.i.d. random variables? Is $M_n$ a random variable? If yes how can we describe it with the $Z_i$? Does $M_n/n$ converge as $n → ∞$ ? Does $n^{−1/2}(M_n − P_n)$ converge in law as $n → ∞$ ? thanks for the help
As a partial answer to get you started. You can describe $M_n$ and $P_n$ as sums of iid indicator random variables: For instance, if we set $I_i = \cases{1 {\text { if } } Z_i\le k\\0{\text { otherwise } }}$, then $M_n = I_1 + \cdots + I_n$. This is a common construction, for example the binomial distribution is a sum of iid Bernoulli random variables, which each itself can be thought as an indicator random variable. We have $M_n/n$ converge in probability to the expected value of $I_i$, which is $E(I_i)=P(Z_i\le k)$. See weak law of large numbers (https://en.wikipedia.org/wiki/Law_of_large_numbers).
Also note, $M_n + P_n = n$, so $M_n - P_n = n- 2P_n$, this may be helpful to analyze your other question.