Editing Convergence of random variables (section)

===Properties===
* Convergence in probability implies convergence in distribution.<sup>[[Proofs of convergence of random variables#propA2|[proof]]]</sup>
* In the opposite direction, convergence in distribution implies convergence in probability when the limiting random variable ''X'' is a constant.<sup>[[Proofs of convergence of random variables#propB1|[proof]]]</sup>
* Convergence in probability does not imply almost sure convergence.<sup>[[Proofs of convergence of random variables#propA1i|[proof]]]</sup>
* The [[continuous mapping theorem]] states that for every continuous function <math>g</math>, if <math display="inline">X_n \xrightarrow{p} X</math>, then also&thinsp;{{nowrap|<math display="inline">g(X_n)\xrightarrow{p}g(X)</math>.}}
* Convergence in probability defines a [[topology]] on the space of random variables over a fixed probability space. This topology is [[metrizable]] by the ''[[Ky Fan]] metric'':<ref>{{harvnb|Dudley|2002|page=289}}</ref> <math style="position:relative;top:.3em" display="block">d(X,Y) = \inf\!\big\{ \varepsilon>0:\ \mathbb{P}\big(|X-Y|>\varepsilon\big)\leq\varepsilon\big\}</math> or alternately by this metric <math display="block">d(X,Y)=\mathbb E\left[\min(|X-Y|, 1)\right].</math>