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Set $M_n={\sum }_{j=0}^n{X}_j$ and ${\mathcal{F}}_n=\sigma (X_0,\dots ,X_n)$. The independence and zero expected values give $\mathbb{E}[M_{n+1}\mid {\mathcal{F}}_n]=M_n$, so $(M_n)$ is a martingale. The variance additivity for independent random variables givesThe Cauchy-Schwarz inequality implies ${\sup }_n\mathbb{E}|M_n|\le \sqrt{V}$. Applying the Martingale convergence theorem from (b),In fact the L2 martingale convergence theorem also gives convergence in L2. Independently, for $n>m$ the same variance additivity for independent random variables yields $\mathbb{E}|M_n-M_m{|}^2={\sum }_{j=m+1}^n\mathbb{E}[X_j^2]\to 0$, confirming that the partial sums form a Cauchy sequence in $L^2$.

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.