Independence among random vectors

Question

Suppose $x_1,x_2$ are independent and identically distributed d- dimensional random vectors.

Let$ x_1 = [ A_1 , A_2, A_3,......A_d ]$

$ X_2 =[B_1,B_2,B_3... B_d] $

Where $A_i, B_i$ are random variables following std. normal distri$b^n$.

Here is my my question , What do you mean by independence ? Are $A_1, A_2 ...A_i$ independent among them or is it the pairs such as $(A_1 ,B_1)(A_2,B_2)....(A_n, B_n) $ independent?

Answer 1

Independence means that information about the value of one variable doesn't inform you about the probability distribution of the other. hus $x_1$ is independent of $x_2$ iff each $x_{2i}$ is independent of each $x_{1i}$, i.e. $B_i$ is independent of $A_i$.