Let $X$ be a random variable which takes on values from $Omega subset mathbb{R}^m$. Assume $Omega$ has a Lipschitz boundary. Let $p(x)$ be the probability density function of $X$ and assume $p(x)>0forall xinOmega$.

Assume we draw iid, two sets of n points each of the variable $X$, from the pdf $p$. Let them be denoted as $A = {p_1,p_2ldots p_n}$ and $B = {q_1,q_2ldots q_n}$.

Consider $zeta_n = maxlimits_{xin B}minlimits_{yin A} |x-y|_2$

I’d like to know the decay rate of $zeta_n$.

Also how does it converge to zero? (in probability? or in expectation?)

What I know till now?

Consider $gamma_n = suplimits_{xin Omega}minlimits_{yin A} |x-y|_2$

I think $gamma_n sim n^{-frac{1}{m}}$.

I’d like to know if we can say similar thing about $zeta_n$?