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Let $X$ be a random variable with values in a compact $\Omega \subset \mathbb{R}^m$. Assume $\Omega$ is compact and has a Lipschitz boundary. Let $p(x)$ be the probability density function of $X$ and assume $p(x)>0\ \forall x\in\Omega$.

Now we draw two independent sets of $n$ iid copies of $X$. We denote them $A_n = \{p_1,p_2\ldots p_n\}$ and $B_n = \{q_1,q_2\ldots q_n\}$. Consider $$\ \ \ \ \zeta_n = \max\limits_{x\in B_n}(dist(x,A_n)) = \max\limits_{x\in B_n}\min\limits_{y\in A_n} \|x-y\|_2$$

Questions:

1. Does $\zeta_n$ converge to zero, in probability or in expectation?

2. What can we say about the rate of decay of $\zeta_n$? If we consider $$\gamma_n = \sup\limits_{x\in \Omega}(dist(x,A_n)) = \sup\limits_{x\in \Omega}\min\limits_{y\in A} \|x-y\|_2$$ instead, then I think $\gamma_n \sim n^{-1/m}$. Can we say anything similar thing about $\zeta_n$?

Let $X$ be a random variable with values in a closed compact $\Omega \subset \mathbb{R}^m$. Assume $\Omega$ is has a Lipschitz boundary. Let $p(x)$ be the probability density function of $X$ and assume $p(x)>0\ \forall x\in\Omega$.

Now we draw two independent sets of $n$ iid copies of $X$. We denote them $A_n = \{p_1,p_2\ldots p_n\}$ and $B_n = \{q_1,q_2\ldots q_n\}$. Consider $$\ \ \ \ \zeta_n = \max\limits_{x\in B_n}(dist(x,A_n)) = \max\limits_{x\in B_n}\min\limits_{y\in A_n} \|x-y\|_2$$

Questions:

1. Does $\zeta_n$ converge to zero, in probability or in expectation?

2. What can we say about the rate of decay of $\zeta_n$? If we consider $$\gamma_n = \sup\limits_{x\in \Omega}(dist(x,A_n)) = \sup\limits_{x\in \Omega}\min\limits_{y\in A} \|x-y\|_2$$ instead, then I think $\gamma_n \sim n^{-1/m}$. Can we say anything similar thing about $\zeta_n$?

Let $X$ be a random variable with values in a compact $\Omega \subset \mathbb{R}^m$. Assume $\Omega$ has a Lipschitz boundary. Let $p(x)$ be the probability density function of $X$ and assume $p(x)>0\ \forall x\in\Omega$.

Now we draw two independent sets of $n$ iid copies of $X$. We denote them $A_n = \{p_1,p_2\ldots p_n\}$ and $B_n = \{q_1,q_2\ldots q_n\}$. Consider $$\ \ \ \ \zeta_n = \max\limits_{x\in B_n}(dist(x,A_n)) = \max\limits_{x\in B_n}\min\limits_{y\in A_n} \|x-y\|_2$$

Questions:

1. Does $\zeta_n$ converge to zero, in probability or in expectation?

2. What can we say about the rate of decay of $\zeta_n$? If we consider $$\gamma_n = \sup\limits_{x\in \Omega}(dist(x,A_n)) = \sup\limits_{x\in \Omega}\min\limits_{y\in A} \|x-y\|_2$$ instead, then I think $\gamma_n \sim n^{-1/m}$. Can we say anything similar thing about $\zeta_n$?

Let $X$ be a random variable with values in a compact $\Omega \subset \mathbb{R}^m$. Assume $\Omega$ is compact and has a Lipschitz boundary. Let $p(x)$ be the probability density function of $X$ and assume $p(x)>0\ \forall x\in\Omega$.

Now we draw two independent sets of $n$ iid copies of $X$. We denote them $A_n = \{p_1,p_2\ldots p_n\}$ and $B_n = \{q_1,q_2\ldots q_n\}$. Consider $$\ \ \ \ \zeta_n = \max\limits_{x\in B_n}(dist(x,A_n)) = \max\limits_{x\in B_n}\min\limits_{y\in A_n} \|x-y\|_2$$

Questions:

1. Does $\zeta_n$ converge to zero, in probability or in expectation?

2. What can we say about the rate of decay of $\zeta_n$? If we consider $$\gamma_n = \sup\limits_{x\in \Omega}(dist(x,A_n)) = \sup\limits_{x\in \Omega}\min\limits_{y\in A} \|x-y\|_2$$ instead, then I think $\gamma_n \sim n^{-1/m}$. Can we say anything similar thing about $\zeta_n$?

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

Now we draw two independent sets of $n$ iid copies of $X$. We denote them $A_n = \{p_1,p_2\ldots p_n\}$ and $B_n = \{q_1,q_2\ldots q_n\}$. Consider $$\ \ \ \ \zeta_n = \max\limits_{x\in B_n}(dist(x,A_n)) = \max\limits_{x\in B_n}\min\limits_{y\in A_n} \|x-y\|_2$$

Questions:

1. Does $\zeta_n$ converge to zero, in probability or in expectation?

2. What can we say about the rate of decay of $\zeta_n$? If we consider $$\gamma_n = \sup\limits_{x\in \Omega}(dist(x,A_n)) = \sup\limits_{x\in \Omega}\min\limits_{y\in A} \|x-y\|_2$$ instead, then I think $\gamma_n \sim n^{-1/m}$. Can we say anything similar thing about $\zeta_n$?

Let $X$ be a random variable with values in a compact $\Omega \subset \mathbb{R}^m$. Assume $\Omega$ has a Lipschitz boundary. Let $p(x)$ be the probability density function of $X$ and assume $p(x)>0\ \forall x\in\Omega$.

Now we draw two independent sets of $n$ iid copies of $X$. We denote them $A_n = \{p_1,p_2\ldots p_n\}$ and $B_n = \{q_1,q_2\ldots q_n\}$. Consider $$\ \ \ \ \zeta_n = \max\limits_{x\in B_n}(dist(x,A_n)) = \max\limits_{x\in B_n}\min\limits_{y\in A_n} \|x-y\|_2$$

Questions:

1. Does $\zeta_n$ converge to zero, in probability or in expectation?

2. What can we say about the rate of decay of $\zeta_n$? If we consider $$\gamma_n = \sup\limits_{x\in \Omega}(dist(x,A_n)) = \sup\limits_{x\in \Omega}\min\limits_{y\in A} \|x-y\|_2$$ instead, then I think $\gamma_n \sim n^{-1/m}$. Can we say anything similar thing about $\zeta_n$?