Uniformities for the convergence in law and in probability (Q1185794)

For random variables \(X\), \(Y\) taking values in a complete separable metric space \((S,d)\), the Ky-Fan metric is defined by \[ K(X,Y)=\inf\{\varepsilon>0:\;P(d(X,Y)>\varepsilon)<\varepsilon\}. \] Convergence in probability is metrized by \(K\). The authors show that for \(S\)-valued random variables \((X_ n,Y_ n)\), \(K(X_ n,Y_ n)\to 0\) is equivalent to the existence of random variables \((\tilde X_ n,\tilde Y_ n)\) on some probability space with the same distribution as \((X_ n,Y_ n)\) and with \(d(\tilde X_ n,\tilde Y_ n)\to 0\) a.s.. The question of when \(K(X_ n,Y_ n)\to 0\) is equivalent to \(d_ F(X_ n,Y_ n)=\sup_{f\in F}E| f(X_ n)-f(Y_ n)|\to 0\) for suitable classes of functions \(F\) (bounded Lipschitz) is considered. Similar questions for metrics metrizing convergence in distribution are also addressed.