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Let $X$ and $Y$ be independent random vectors (denote this by $X \perp Y$ ) . Let $A=[|X|\leq t ]$ and $B=[|Y|\leq t ]$. Since $X\perp Y$, we know that $A$ and $B$ are independent, so the indicators $\mathbf{1}_A$ and $\mathbf{1}_B$ are independent.

I wanto to show that $X \mathbf{1}_A$ and $Y \mathbf{1}_B$ are independent.

My strategy is to show a more general fact: if $X \perp Y$ and $X_1 \perp Y_1$, then $XX_1 \perp Y Y_1$. Could you give me some hint to show this? If this is not true, can you give me one hint to show my original question?

André Goulart
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    What are the subscripts on the $X$ and $Y$ in your last paragraph indicative of? If they are just other variables, the generic statement isn't true; consider $X_1 = Y$ and $Y_1 = X$, then $XX_1 = YY_1$. – jbowman Jul 25 '23 at 00:38
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    Generically, though, independence means that $P(X,Y) = P(Y)P(X) ,\forall ,X,Y$, so it clearly must hold on any subset of $X,Y$ as well. – jbowman Jul 25 '23 at 00:41

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