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Let independency of two random variables $A$ and $B$ be defined as $P(A,B) = P(A)P(B)$.

Let's consider two vectors as variables $\mathbf{X} = (X_1, X_2)$ and $\mathbf{Y} = (Y_1,Y_2)$.

Must we have that $\mathbf{X}$ and $\mathbf{Y}$ are independent if the elements $X_i$ and $Y_j$ are independent for all $i$ and $j$?

Related: In Intuition on Independence of Random Vectors , it is explained that independence of vectors $\mathbf{X}$ and $\mathbf{Y}$ implies independence of $X_i$ and $Y_j$. Is the reverse also true?

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The answer is no.

An example is for a vector $X$ of size 1 and $Y$ of size 2.

Let $X$ be Bernoulli variable with equal probability. Let $Y$ be distributed, with equal probability, among $(1,1)$ or $(0,0)$ if $X=0$ and among $(1,0)$ and $(0,1)$ if $X=1$.

Then following observations will occur with equal probability

X1  Y1   Y2
1    1    0
1    0    1
0    1    1
0    0    0

Here $X_1$ is independent from $Y_1$ and $X_1$ is independent from $Y_2$. But $X_1$ is not independent from $(Y_1,Y_2)$.

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