Suppose that $(X_t)$ and $(Y_t)$ are stochastic processes defined on the same probability space whose sample paths belong to some Hilbert space $K$ (or more generally, to some function space). We may view these processes as $K$-valued random variables, hence we may talk about their independence as random variables.
Is the independence of stochastic processes $(X_t)$ and $(Y_t)$ equivalent to the independence of the corresponding $K$-valued random variables?
Apologies if this is trivial, but I am lost with indices.
I am not sure, if I understood this question correctly. The way I understood it, as we need a measure for independence, we first must think about the natural measure on $\Omega\times I$, where I is the index set of the processes corresponding to the domain of the function space. In my opinion, this would lead again to the finite dimensional distributions and thus to equivalence.