# Independence of random vectors

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?

• Apparently, not. math.stackexchange.com/questions/1006256/… Apr 23, 2018 at 10:03
• 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/….
– whuber
Apr 23, 2018 at 14:25
• @Greenparker Will it hold true if X and Y are multivariate normal distribution? Apr 23, 2018 at 15:15
• Yes--and that is explicitly pointed out in the reference given by @Greenparker.
– whuber
Apr 24, 2018 at 14:27