# 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?