Assume that $X_n \rightarrow X$ in distribution. Are there $Y_n$ and $Y$ such that $Y_n \rightarrow Y$ almost surely and their distribution functions satisfy $F_{X_{n}}=F_{Y_{n}}$, $F_X=F_Y$?
2026-04-14 03:32:31.1776137551
About distribution convergence consequence
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Yes. This is called the Skorokhod representation theorem. Note that, using the notation from wikipedia, separability of the support of $\mu_\infty$ is always given when the metric space $S$ is already separable - as is the case for $S=\Bbb R^N$.