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Consider a sequence of stochastic processes $X_n$ on $L^\infty([0,1]^d)$ and assume that $\mathscr{L}(X_n)$ converges weakly to $\mathscr{L}(W)$, where $W$ is a Brownian bridge and $\mathscr{L}(\cdot)$ denotes the probability law of the random object "$\cdot$".

QUESTION 1: does this also implies weak convergence on a larger space, say $D([0,1]^d)$ - which extends the usual Skohorod's space of càdlàg functions, where suprema are measurable?

QUESTION 2: if the answer to the first question is yes, can we then automatically conclude that $\mathscr{L}(\sup_{t \in [0,1]^d} X_n(t))$ converges weakly to $\mathscr{L}(\sup_{t \in [0,1]^d} W(t))$ in $\mathbb{R}$?

I guess answers to this questions are known in the literature: could you point me to references? E.g., I could not find much on Billingsley's book on convergence of probability measures, except in the case $d=1$.

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