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 $p(x)>0\forall x\in\Omega$.

Assume we draw iid, two sets of n points each of the variable $X$, from the pdf $p$. Let them be denoted as $A = \{p_1,p_2\ldots p_n\}$ and $B = \{q_1,q_2\ldots q_n\}$.

Consider $\zeta_n = \max\limits_{x\in B}\min\limits_{y\in A} \|x-y\|_2$

 I'd like to know the decay rate of $\zeta_n$.
Also how does it converge to zero? (in probability? or in expectation?)

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What I know till now?

Consider $\gamma_n = \sup\limits_{x\in \Omega}\min\limits_{y\in A} \|x-y\|_2$

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

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