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Uniform tightness plus weak convergence of the finite dimensional distributions is sufficient for a stochastic processes $X_n(t)$ to converge in law in spaces like the Donsker space $D[0,1]$ or the space of continuous functions $C[0,1]$, all equipped with a proper metric.

Generally: When is tightness plus weak convergence of the finite dimensional distributions enough to conclude that a Stochastic process converges in distribution to a limiting process?

Specifically: I was wondering about the case of convergence in the space of $\ell^{\infty}(\mathbb R)$, the space of bounded real valued functions equipped with the $\lVert\cdot\rVert_{\infty}$ - norm.

It's interesting, some books, like Klenke "Probability theory", only introduce convergence of the fdd in a continuous path space, so not even in $D[0,1]$, some references only deal with the Donsker space.

Is there a general result - something like: In any metric path space it is sufficient to show the weak convergence of the finite dimensional distributions and the uniform tightness of the distributions? Is there any reference, maybe specifically for the $\ell^{\infty}(\mathbb R)$ case?

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