Let $(X,Y)$ be a pair of random variables on a measure space $\mathcal T \subseteq \text{"subsets of }\mathbb R^2\text{"}$, with joint probability distribution $P$.

We don't assume $X$ and $Y$ are independent!

Let $P_X$ (resp. $P_Y$) be the marginal distribution of $X$ (resp. $Y$), defined by $P_X(A) := \int_{p_1(A)} \,dP$, where $\mathcal p_k(A) := \{(z_1,z_2) \in \mathcal T \mid z_k \in A\}$ defines the projection operator unto the $k$ coordinate of $\mathcal T$. Let $\Pi(X,Y)$ be the set of all couplings of $X$ and $Y$, i.e the set of all probability distributions on $\mathcal T$, with same margins as $P$. Finally, let $P_X \otimes P_Y \in \Pi(X,Y)$ be the independence coupling of $X$ and $Y$ defined by $$ (P_X \otimes P_Y)(U) := P_X(p_1(U))\cdot P_Y(p_2(U)). $$

Let $k$ and $n$ be positive integers, presumably, with $n \gg k$.

Question.Given $n$ independent copies $(X_1,Y_1),\ldots,(X_n,Y_n)$ of $(X,Y)$ (i.e an iid sample of size $n$ from the joint distribution $P$), what is a principled way to obtain an iid sample from $P_X \otimes P_Y$ of size $k$ ?

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