Definition (Random Vector)

Let $X_1,...,X_n$ be $n$ random variables. The vector $X=(X_1,..,X_n)^T$ is called a random vector. So we define $X:S\to\mathbb{R}^n$ such that $(S,p)$ is the underlying probability space.

Definition (Random Vector)

A random vector $Z=(X_{1},\dots,X_{n})$ is a Borel function from $(\Omega,\mathcal{F},\mathbb{P})$ to $\mathbb{R}^{n}$.