Given a random function with parametric form, how do we know if it can be described as a GP? For example, let $$f(t) = t^2 + X$$ where $X \sim \mathcal{N}(0,1)$. Then $f(t)$ is a random function, where each realization has a normal distribution: $f(t) \sim \mathcal{N}(t^2, 1)$. I know that the marginal being Gaussian doesn't imply the joint being Gaussian, so it's not clear to me how to describe the distribution of $f$. If it's not a GP, then how could we describe the (joint) distribution?
The more specific case I'm thinking about is of the form $$ F(x) = \sum_{i=1}^n f_i(x)$$ where $f_i(x)$ are iid from some distribution over functions. Each realization is also asymptotically normal but the central limit theorem, so I was thinking the above case would shed insight here.