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I have recently learnt about kernels in machine learning. And I have been introduced to many different processes e.g. Gaussian process, Wiener process. Now my question is why a set of functions has been named as a process? For example this is the definition of Gaussian process:

Let $\mu: X \to R$ be any function, $k: X \times X\to R$ be a Mercer kernel. A Gaussian process $p(f) = GP(f;μ,k)$ is a probability distribution over the function $f : X \to R$, such that every finite restriction to function values $f_X :=[f_{x_1},...,f_{x_N}]$ is a Gaussian distribution $p(f_X)=N(f_X;μ_X,k_{XX})$.

Which parts exactly associates to a process??

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I believe it's called a process because it takes place over and can change over time. In the case of your examples, they are stochastic processes.

EDIT: As Benjamin says in a comment, the requirement isn't time, but that there is an indexing set, which could be time. It could also be spatial.

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    $\begingroup$ a process does not necessarily have to take place over/change over time, but there is some set that 'indexes' a collection of random variables. This set can be continuous (e.g. time) or discrete (e.g. a set of possible states like the 4 possible nucleotides in statistical genetics). $\endgroup$ – bdeonovic Feb 9 '14 at 20:17

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