regarding the concept of infinite long vector and function

Question

When learning the Gaussian process, there is a concept stating that " infinite long vector is similar to function", which is shown in the attached image. What does it mean, I am not very clear about this statement.

Answer 1

Let $X$ be a vector that follows an $n$-dimensional Gaussian distribution with mean vector $\mu\in\mathbb{R}^n$ and a suitable $n\times n$ covariance matrix $\Sigma$. So, if we index the vector by the numbers $I:=\{1,2.\ldots,n\}$, we could also think about this as a function $I\to\mathbb{R},\,i\mapsto \mu_i$ that maps the $n$ indices to the corresponding means. Similarly, the covariance matrix could be thought of as a function $I\times I\to\mathbb{R},\, (i,j)\to \Sigma_{i,j}$ that maps the $n^2$ pairs of indices to the corresponding covariances.

Now, let's move on to the Gaussian process, which is a family of random variables $(X_t)_{t\in T}$ for some infinite index set $T$ (often $T=\mathbb{N}$ or $\mathbb{R}_+$). Now, we have infinitely many means. So for each $t\in T$ we have a mean $\mu_t$. So, again we can think about this as a function $T\to\mathbb{R},\,t\mapsto \mu_t$. Similarly, for each of the infinitely many pairs of indices $(s,t)$ we have a covariance $\sigma_{s,t}>0$, i.e., a function $T\times T\to\mathbb{R},\, (s,t)\mapsto \sigma_{s,t}$.

In conclusion, even in the finite case, one can think of it as functions, but we do not need to because we can also think about it as vectors/matrices. In the infinite case, vectors/matrices won't work and we have to think about it as functions.