Independence of stochastic processes

Question

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.

Why the downvote? Isn't it a legitmate question? – user512365 Jan 21 '18 at 18:22 — user512365, Jan 21 '18 at 18:22
@flebool, I've already linked your question to mine. – user512365 Jan 21 '18 at 19:23 — user512365, Jan 21 '18 at 19:23

score 0 · Answer 1 · answered Mar 09 '20 at 11:31

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.

Independence of stochastic processes

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