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Suppose first thatThis is convergence in L2, hence convergence in probability, which implies convergence in distribution. By part (i), the respective laws are ${\mu }_n$ and $\nu$, so ${\mu }_n$ converges weakly to $\nu$. Part (a)(i) also gives

Conversely, suppose that ${\mu }_n$ converges weakly to $\nu$ and the displayed second moments converge. By the stated quantile coupling for convergence in distribution,Part (i) identifies their second moments with those of the measures, so part (a)(ii) applies and gives convergence in $L^2$. ThereforeThis is the one-dimensional second-Wasserstein convergence criterion realized by the common-quantile coupling.

Let $F^{-1}(u)=\inf \{x:u\le F(x)\}$. The generalized-inverse identitygivesand similarly $\mathbb{P}({\tilde{X}}_n\le x)=F_{X_n}(x)$. Thus the coupled variables have the required marginal distributions.

At every continuity point $u$ of the nondecreasing function $F_X^{-1}$, convergence of the distribution functions at continuity points of $F_X$ impliesIndeed, values just below and above $F_X^{-1}(u)$ can be chosen at continuity points of $F_X$ and trap the two generalized inverses. A monotone function has at most countably many discontinuities, so this convergence holds for Lebesgue-almost every $u\in (0,1)$. Therefore ${\tilde{X}}_n\to \tilde{X}$ almost surely. This is the quantile coupling for convergence in distribution.