Independence of random vectors if the coordinates are independent

Question

Suppose that $X_n$ is $n$ independent copies of $X$ and $Y_n$ is n independent copies of $Y$. Define $Z_i = (X_i, Y_i)$. What can I say about $Z_1, Z_2, \dots Z_n$. Are they mutually independent.

Answer 1

Let it be that $X,Y,U,V$ are iid.

Setting $X_1=U$ and $X_2=V$ we have $2$ independent copies of $X$.

Setting $Y_1=V$ and $Y_2=U$ we have $2$ independent copies of $Y$.

But $Z_1=(X_1,Y_1)=(U,V)$ and $Z_2=(X_2,Y_2)=(V,U)$ are not necessarily independent.