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Intuitively, two random variables $X$ and $Y$ are independent if knowing the value of one of the random variable provides zero information about the other. The same holds true for two random vectors $\mathbf{X}=(X_1,X_2,\cdots, X_m), \mathbf{Y}=(Y_1,Y_2,\cdots, Y_n)$. But does it also mean that $\mathbf{X}$ and $\mathbf{Y}$ are independent componentwise? I mean, is $X_i$ independent of $Y_j$ for every $1\le i\le m$ and $1\le j\le n$?

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Yes, $X_i$ is independent of $Y_j$. To see this, note that if $\mathbf{X}$ and $\mathbf{Y}$ are independent, then for functions $f$ and $g$, $f(\mathbf{X})$ and $g(\mathbf{Y})$ are independent. See discussion here for this statement.

So let $f$ be the function that picks out the $i$th element of $\mathbf{X}$, that is, $f(\mathbf{X}) = X_i$ and similarly define $g(\mathbf{Y}) = Y_j$. Then $X_i$ is independent of $Y_j$.

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