About: Proofs of convergence of random variables

This article is supplemental for “Convergence of random variables” and provides proofs for selected results. Several results will be established using the portmanteau lemma: A sequence {Xn} converges in distribution to X if and only if any of the following conditions are met: 1. * E[f(Xn)] → E[f(X)] for all bounded, continuous functions f; 2. * E[f(Xn)] → E[f(X)] for all bounded, Lipschitz functions f; 3. * limsup{Pr(Xn ∈ C)} ≤ Pr(X ∈ C) for all closed sets C;