why do we need a sequence of random variable, isn't one function sufficient?

Question

In sampling, we have so many situations involving a sequence of random variables, what I am confusing is why do we need a sequence of random variables to describe the process? It feels like each function is only used once.

Suppose $$X_i:\Omega\to\mathbb{R}\quad,i\in\mathbb{N}$$ $X_1,X_2,X_3...$ basically just the $\mathbb{R}$-valued image, each image has its corresponding function.

why don't we only use a single random variable to describe those images, this also seems to be sufficient to describe the process, if not, what is the problem?

Answer 1

In the standard framework, a single outcome $\omega \in \Omega$ in a probability space determines the values of all random variables $X_1(\omega), X_2(\omega), ...$ that you have defined on the system. But your idea of "repeating the experiment" and "using the same function" can be done within this framework using the product space:

Suppose you start with a (small) probability space $(\Omega, \mathcal{F}, P)$ on which lies a random variable $X:\Omega\rightarrow\mathbb{R}$ that has a Gaussian $N(0,1)$ CDF.

One way to "repeat the experiment an infinite number of times" but to maintain our correct view of probability is to define a new (large) probability space $(\tilde{\Omega}, \tilde{\mathcal{F}}, \tilde{P})$ that is big enough to fit everything we want:

\begin{align} &\tilde{\Omega} = \Omega \times \Omega \times \Omega \times ...\\ &\tilde{\mathcal{F}} = \mathcal{F}\otimes \mathcal{F} \otimes \mathcal{F} \otimes. ... \end{align} and where our new outcomes $\tilde{\omega} \in \tilde{\Omega}$ have the form: $$\tilde{\omega} = (\omega_1, \omega_2, \omega_3, ...) $$ where $\omega_i \in \Omega$ for all $i\in \{1, 2, 3, ...\}$. Now the new probability measure $\tilde{P}:\tilde{\mathcal{F}}\rightarrow\mathbb{R}$ is constructed from the old one $P:\mathcal{F}\rightarrow \mathbb{R}$ as the unique measure that satisfies \begin{align} &\tilde{P}[\{\tilde{\omega} \in \tilde{\Omega} : \omega_1 \in A_1, \omega_2 \in A_2, \ldots, \omega_n \in A_n\}] \\ &= \prod_{i=1}^n P[A_i] \end{align} for all positive integers $n$ and all $A_1, ..., A_n \in \mathcal{F}$. This new probability space $(\tilde{\Omega}, \tilde{\mathcal{F}}, \tilde{P})$ is called the product space.

In this case you can actually define i.i.d. $N(0,1)$ random variables $X_i:\tilde{\Omega}\rightarrow\mathbb{R}$ using the same original function $X:\Omega\rightarrow\mathbb{R}$ by $$X_i(\tilde{\omega}) = X(\omega_i) \quad \forall \tilde{\omega} \in \tilde{\Omega}, \forall i \in \{1, 2, 3, ...\}, $$ That is, for all $\omega_i\in \Omega$ and $i \in \{1, 2, 3, ...\}$: $$\boxed{X_i(\omega_1, \omega_2, \omega_3, ...) = X(\omega_i)} $$ For this large probability space $(\tilde{\Omega}, \tilde{\mathcal{F}}, \tilde{P})$, the function $X:\Omega\rightarrow \mathbb{R}$ is just a function that is used to construct the random variables $X_i$. The function $X:\Omega\rightarrow\mathbb{R}$ can no longer can be viewed as a "random variable" because it is not defined on $\tilde{\Omega}$.

Answer 2

I don't see how this would make any difference.

I think it's mostly due to notation and history. The notation $a_i$, used to indicate a sequence of concrete values, was already around, and when people were formalizing random variables they just generalized the concept. Where as I'm not familiar with any notation for "the first value generated from random variable $X$", "the second value generated from random variable $X$", ... "the $n$th value generated from random variable $X$".

I wonder if you're coming from a computer programming background, where there is some cost to create a "random variable", and so it seems wasteful to use it once and then throw it away. Also, computer programming is (generally) sequential, so you can only generate one sample at a time. But in math you can just as easily think of a whole sequence of random variable, and "generate" a single sample from all of them at once. Additionally, the idea that you "generate" values doesn't even really exist in the mathematical formalism - we can just as well think of the values as "already being there", but that they have a particular distribution.