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Let $\mathbf{X}=(X_1,X_2,\cdots, X_n)$ and $\mathbf{Y}=(Y_1,Y_2,\cdots, Y_m)$ be two random vectors. If each component of $\mathbf{X}$ is independent of $\mathbf{Y}$ can we say that $\mathbf{X}$ and $\mathbf{Y}$ are independent? In other words, if $X_i$ is independent of $Y_j$ for every $1\le i \le n$ and $1\le j\le m$ then are $\mathbf{X}$ and $\mathbf{Y}$ independent? If not, what about the special case when $\mathbf{X}$ and $\mathbf{Y}$ are Multivariate Normals?

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    $\begingroup$ Apparently, not. math.stackexchange.com/questions/1006256/… $\endgroup$ Apr 23 '18 at 10:03
  • $\begingroup$ This is very closely related to the possibility that three variables might fail to be independent while any pair of them is independent: stats.stackexchange.com/questions/51322/…. $\endgroup$
    – whuber
    Apr 23 '18 at 14:25
  • $\begingroup$ @Greenparker Will it hold true if X and Y are multivariate normal distribution? $\endgroup$ Apr 23 '18 at 15:15
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    $\begingroup$ Yes--and that is explicitly pointed out in the reference given by @Greenparker. $\endgroup$
    – whuber
    Apr 24 '18 at 14:27

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