# Two definitions of a Stochastic Process?

I have two supposedly equivalent definitions of a stochastic process.

1. A stochastic process is an indexed set of random variables. Specifically $$y = \{y(x) \; | \; x \in \mathcal{X}\}.$$ Typically $$y \in \mathbb{R}$$ and $$\mathcal{X} \subset \mathbb{R}^n$$. If the index set is infinite, then we might say that the stochastic process is continuous.
2. A stochastic process is a probability distribution over a space of functions.

I'm trying to read through Gaussian Processes for Machine Learning by Rasmussen and Williams, and they mention that you can consider a Gaussian Process to be a distribution over a space of functions.

Question: I have the definition that a Gaussian process is a stochastic process (in the sense of definition 1.) for which every finite collection of random variables has the property that $$(y(x_1),...,y(x_n)) \sim \mathcal{N}(\mathbf{\mu},\mathbf{\Sigma}), (\forall \; n \in \mathbb{N} \text{ and } x_1,...,x_n).$$ How can I reconcile this definition with the perspective from definition (2)? I guess that I should be thinking about $$y$$ as being a function so $$y(\cdot)$$ is more appropriate, but then is $$y(\cdot)$$ a collection of random functions? How does it define a distribution, and can it be written down explicitly? I hope my questions are clear, I'm new to the material with minimal background knowledge. Thanks in advance for the help!

In the definition you start with, $$\tag{1} (y(x_1),...,y(x_n)) \sim \mathcal{N}(\mathbf{\mu}_n,\mathbf{\Sigma}_n),\quad (\forall \; n \in \mathbb{N} \text{ and } x_1,...,x_n).$$ we need labels at the mean vector $$\mu_n$$ and at the covariance matrix $$\Sigma_n$$.
The index set of this process is the space the $$x_i$$ live in. It seems you mean $$x_i\in\mathbb R^\color{red}{d}$$ for some $$\color{red}{d}$$. So $${\cal X}\subset\mathbb R^d$$. (We should not use $$n$$ twice.) Further it is clear that $$y(x_i)\in\mathbb R$$.
We can now look at $$y$$ as being a function from $${\cal X}$$ to $$\mathbb R$$ and definition (1) says that for each $$n$$ and every vecor $$(x_1,...,x_n)$$ the random variable $$(y(x_1),...,y(x_n))$$ is $$n$$-variate Gaussian. In other words, the distribution of $$(y(x_1),...,y(x_n))$$ is a finite dimensional Gaussian probability measure on $$\mathbb R^n$$.
Note that $$(x_1,...,x_n)\mapsto (y(x_1),...,y(x_n))$$ is a function from $$\mathbb R^n$$ to $$\mathbb R^n$$ whilst $$y$$ is a function from a subset of $$\mathbb R^\color{red}{d}$$ to $$\mathbb R$$.
The not very trival but intuitively plausible Kolmogorov extension theorem says under what conditions there exists an infinite dimensional Gaussian measure on the set of functions $$y:\mathbb R^\color{red}{d}\to \mathbb R$$ that is compatible with the finite dimensional one above in the sense that (1) holds. Strictly speaking we encounter this theorem mostly in the context of the Wiener process. I hope you are not asking me now to prove it for (1). Getting the intuition is I think more important at that stage.