Question

Should the definition of random vector be associated with P: the probability set function?

Answer 1

The concept of random vectors is a multidimensional generalization of the concept of random variable.

Suppose that we conduct a probabilistic experiment and that the possible outcomes of the experiment are described by sample space $\Omega$ .

A random vector is a vector whose value depends on the outcome of the experiment, as stated by the following definition.

Let $\Omega$ be a sample space. A random vector X is a function from sample space $\Omega$ to the set of K-dimensional real vectors $\mathbb{R}^{\kappa}$ :

$X = \Omega → \mathbb{R}^{\kappa}$

In rigorous probability theory, the function X is also required to be measurable. We report here a more rigorous definition of random vector by using the formulation of measure theory.

Let $(\Omega, F, P)$ be a probability space. Let $\beta (\mathbb{R}^{\kappa})$ be the Borel sigma- algebra of $\mathbb{R}^{\kappa }$ . A function $X : \Omega \rightarrow \mathbb{R}^{\kappa }$ such that $\left \{ \infty \epsilon \Omega :X(\infty ) \epsilon B\right \} \epsilon F \; for \; any \; B \epsilon \beta (\mathbb{R}^{\kappa })$ is said to be a random variable on $\Omega$ .

This definition ensures that the probability that realization of the random vector X will belong to set $B \epsilon \beta (\mathbb{R}^{\kappa})$ can be defined as $P( X \epsilon B) = P({ \infty \epsilon \Omega : X(\infty ) \epsilon B})$ because the set ${ \infty \epsilon \Omega : X(\infty ) \epsilon B}$ belongs to the sigma- algebra F and as a consequence it's probability is well defined.