Difference bewteen random process and functional random variable.

I would like to ask some basic question but I cannot find the appropriate answer on it. What is the basic diference between funcional random variable and random process?

Random process $$\mathbb{X}$$ is defined (for example) as sequence of random variables defined on the same probability sapce $$\left(\Omega, S, P\right)$$, i. e. $$\mathbb{X}=\left\{X_t\right\}_{t\in T}$$, where $$X_t$$ is random variable for each $$t\in T$$. More precisely, we can consider $$\mathbb{X}$$ as function from $$\mathbb{X}(\omega,t):\Omega\times T\to \mathbb{R}$$. When we fix $$\omega$$ we have trajektory of random process. It means that if we consider time $$T=\mathbb{R}$$, we have for fixed $$\omega$$ real function on $$\mathbb{R}$$. On the other hand when we fix $$t\in \mathbb{R}$$ we have ("ordinary") random variable.

Informally in some literature, we can find that functional random "variable" is as random curve (or as random element of $$L^2$$ space).

I still not quite well understand the difference between random process (which is also random curve, in my opinion) and functional random variable.

Probably I little bit understand the difference when I have some realizations. Usualy we have "only one" realization of random process (moreover discrete, but now we are considering continuous realization), it means that we have only one trajectory of random process. Than we try to make for example some predictions and etc.

But in the functional data analysis, we (I think) have more realizations, something like we have more curves with same domain. I consider it as more realizations of some random process.

Can somebody please give me some better and more detailed explanation what is the difference between random process and functional random "variable"?

Any help will be appreciated. Thank you very much.

• In short: a random process is a family (that is usually indexed by a discrete or continuous time) of functional random variables $X_n$ or $X_t$. Commented Aug 8, 2022 at 9:13
• Never heard of a functional random variable. Commented Aug 8, 2022 at 9:15
• Indeed a random process is a time-indexed collection of random variables. When you are talking about curves or other objects that can be random, I wonder if you mean a "random element" (which some texts unfortunately also call a "random variable"). Given a probability space $(\Omega,\mathcal{F}, P)$, a random variable is a measurable function $X:\Omega\rightarrow\mathbb{R}$ while a random element is a measurable function $X:\Omega\rightarrow V$ where $V$ is some nonnempty set and $\mathcal{G}$ is some sigma algebra on $V$. So $V$ can be a set of curves (but it needs a $\mathcal{G}$). Commented Aug 8, 2022 at 11:50
• @ Michael thank you. I understand all what you wrote. But I have troubles to understand what is the main diference between functional data (I think this concept is quite cammon) and realization of stochastic/random process with continuous time. Could you explain me the difference, please? (if you can) Commented Aug 26, 2022 at 7:44

A random process $$\mathbb{X}=(X_t)_{t\in\mathbb{T}}$$ is a family of random variables $$X_t:\Omega\rightarrow\mathbb{R}$$ indexed by an ordered set $$\mathbb{T}$$ e.g. $$\mathbb{R}$$, but more commonly a subset of the nonnegative axis.