May a random-variable be a vector valued function?

Question

Does there exist a random variable X such that $X: \Omega_1 \times \Omega_2 \times ... \times \Omega_p \rightarrow E^q$?

$\Omega_i$ is some set of outcomes.

$E^q$ is q-dimensional measurable space.

Answer 1

You can consider "anything" you like. It's then a matter of terminology. Suppose $(\Omega,\mathcal{A})$ and $(S,\mathcal{B})$ are measurable spaces (i.e sets together with $\sigma$-algebras). You can then consider any $\mathcal{A}$-$\mathcal{B}$-measurable function $X:\Omega\to S$. In probability, one often calls this something like "$S$-valued random object". For example,

• If $S=\Bbb{R}$ and $\mathcal{B}=\mathcal{B}(\Bbb{R})$ is the Borel $\sigma$-algebra, we call $X$ a (real-valued) random variable.
• If $S=\Bbb{R}^k$ and $\mathcal{B}=\mathcal{B}(\Bbb{R}^k)$ is the Borel $\sigma$-algebra on $\Bbb{R}^k$, we often call $X$ a random-vector.
• If $S=M_{m\times n}(\Bbb{R})$ is the space of matrices and $\mathcal{B}$ is the Borel $\sigma$-algebra, we call $X$ a random matrix.
• If $S=\Bbb{R}^{\Bbb{N}}=\text{Functions}(\Bbb{N},\Bbb{R})$ with some $\sigma$-algebra, then one calls $X$ a random sequence (since elements of $S$ are precisely real sequences).
• If for example $S$ is a set of graphs, then one refers to $X$ as a random graph.

And so on. So whatever the elements of the target space $S$ are (reals/complex/ vectors/ Banach-space of functions/ graphs/ sequences etc etc), we call $X$ a random ________ accordingly.