# pr.probability – A problem on rate of decay of fill distance?

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 $$zeta_n$$?