Is there a version of Slutsky theorem for stochastic process? To be more specific, if a stochastic process $X_n(t)$ converges weakly to a tight Gaussian process $G(t)$, and another stochastic process $Y_n(t)$ converges uniformly to a deterministic function $H(t)$, then is it true that $X_n(t)*Y_n(t)$ converges weakly to a tight Gaussian process $H(t)*G(t)$? If so, could you provide a hint or two on how to prove it?
Many thanks!

I think it holds. From the proof of Slutsky's Theorem in Kosorok's book which shows the result when H is a constant instead of a function on t, the key step is that the joint distribution of G(t) and H is completely determined by the marginals. This statement is true when H is a function H(t) but not random.
