# probability distributions – Connection between two types of convergence of random variable

Let $${xi_n}_{n geq 1} : (Omega, mathcal{F}, P) rightarrow (mathbb{R}^1, Bor)$$ be a sequence of r.v.

Please, help me find connections between the following types of convergence as $$n rightarrow infty$$:

1. $$xi_n xrightarrow{sLip} xi$$, i.e. $$Sigma_{n geq 1} E|f(xi_n) – f(xi)| < infty$$ $$forall f in Lip$$ and bounded

2. $$xi_n xrightarrow{sdistr} xi$$, i.e. $$Sigma_{n geq 1} E|F_{xi_n}(x) – F_{xi}(x)| < infty$$ $$forall$$ point $$x$$ of continuity of (distribution) $$F_{xi}$$.

I know that in special cases (for example when $$f$$ is $$arctan$$) the second convergence follows from the first convergence.
But I don’t know how to prove it in general (and find a counterexample that first convergence does not follows from the second, if I understand this correctly).