I have two supposedly equivalent definitions of a stochastic process.
- 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.
- 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!