Let $X=(X_1,X_2)$ and $Y=(Y_1,Y_2)$ be two random vectors in ${\mathrm{R}}^2$. Suppose that $X_i$ and $Y_j$ are independent for each pair of indices. We have the following questions.

1. Are $X$ and $Y$ independent?
2. Are $X$ and $Y$ independent if each of them is bivariately normally distributed?
3. Are $X$ and $Y$ independent if they are jointly normally distributed?

Answer

Let $X_1,X_2\in U(0,1)$ be uniformly distributed and independent. Define the variables

$$\begin{array}{rl}{Y}_1& =X_1+X_2\mathrm{mod}1\\ Y_2& =X_1-X_2\mathrm{mod}1,\end{array}$$

where the $\mathrm{mod}1$ operation means that we identify all integer values on the real line and turn it into a circle of circumference one. Notice that $Y_1\mid X_1\sim U(0,1)$, hence the distribution of $Y_1$ does not depend on $X_1$. By the same token $Y_1$ is independent of $X_2$ as well. The argument also applies to $Y_2$, which is independent of $X_1$ and $X_2$. Hence all pairs of components are independent. However, the knowledge of $X$ fully determines the value of $Y$, hence $X$ and $Y$ are not independent.

We shall answer the second question by reducing it to the first one. To that end, let us consider the variables $X_1\in N(0,1)$ and $X_2\in N(0,1)$, $X_1$ and $X_2$ independent, and apply the transformation

$$\begin{array}{rl}{Z}_1& =F(X_1)+F(X_2)\mathrm{mod}1\\ Z_2& =F(X_1)-F(X_2)\mathrm{mod}1,\end{array}$$

where $F$ is the cumulative distribution function of the standard normal distribution. Since $F(X_i)\in U(0,1)$, it follows that $Z_1$ and $Z_2$ have a standard uniform distribution. We now define the variables $Y_1$ and $Y_2$ such that

$$Y_1|Z_1\sim N(Z_1,1),Y_2|Z_2\sim N(Z_2,1).$$

We conclude that $X_i$ and $Y_j$ are independent for all pair of indices. However, the knowledge of $X$ fully determines the value of $Z$, which is a parameter in the distribution of $Y$. Hence $X$ and $Y$ are not independent.

If $X$ and $Y$ are jointly normally distributed, then pairwise independence does imply vector independence.