Editing Convergence of random variables (section)

===Properties===
* Since <math>F(a) = \mathbb{P}(X \le a)</math>, the convergence in distribution means that the probability for {{mvar|X<sub>n</sub>}} to be in a given range is approximately equal to the probability that the value of {{mvar|X}} is in that range, provided {{mvar|n}} is [[sufficiently large]].
*In general, convergence in distribution does not imply that the sequence of corresponding [[probability density function]]s will also converge. As an example one may consider random variables with densities {{math|''f<sub>n</sub>''(''x'') {{=}} (1 + cos(2''πnx''))'''1'''<sub>(0,1)</sub>}}. These random variables converge in distribution to a uniform ''U''(0,&thinsp;1), whereas their densities do not converge at all.<ref>{{harvnb|Romano|Siegel|1985|loc=Example 5.26}}</ref>
** However, according to ''Scheffé’s theorem'', convergence of the [[probability density function]]s implies convergence in distribution.<ref name="Durrett">{{cite book|last1=Durrett|first1=Rick|title=Probability: Theory and Examples|date=2010|page=84}}</ref>
* The [[portmanteau lemma]] provides several equivalent definitions of convergence in distribution. Although these definitions are less intuitive, they are used to prove a number of statistical theorems. The lemma states that {{math|{''X<sub>n</sub>''} }} converges in distribution to {{mvar|X}} if and only if any of the following statements are true:<ref>{{harvnb|van der Vaart|1998|loc=Lemma 2.2}}</ref>
** <math>\mathbb{P}(X_n \le x) \to \mathbb{P}(X \le x)</math> for all continuity points of <math>x\mapsto \mathbb{P}(X \le x)</math>;
** <math>\mathbb{E}f(X_n) \to \mathbb{E}f(X)</math> for all [[Bounded function|bounded]], [[continuous function]]s <math>f</math> (where <math>\mathbb{E}</math> denotes the [[expected value]] operator);
** <math>\mathbb{E}f(X_n) \to \mathbb{E}f(X)</math> for all bounded, [[Lipschitz function]]s <math>f</math>;
** <math>\lim\inf \mathbb{E}f(X_n) \ge \mathbb{E}f(X)</math> for all nonnegative, continuous functions <math>f</math>;
** <math>\lim\inf \mathbb{P}(X_n \in G) \ge \mathbb{P}(X \in G)</math> for every [[open set]] <math>G</math>;
** <math>\lim\sup \mathbb{P}(X_n \in F) \le \mathbb{P}(X \in F)</math> for every [[closed set]] <math>F</math>;
** <math>\mathbb{P}(X_n \in B) \to \mathbb{P}(X \in B)</math> for all [[continuity set]]s <math>B</math> of random variable <math>X</math>;
** <math>\limsup \mathbb{E}f(X_n) \le \mathbb{E}f(X)</math> for every [[upper semi-continuous]] function <math>f</math> bounded above;{{citation needed|date=February 2013}}
** <math>\liminf \mathbb{E}f(X_n) \ge \mathbb{E}f(X)</math> for every [[lower semi-continuous]] function <math>f</math> bounded below.{{citation needed|date=February 2013}}
* The [[continuous mapping theorem]] states that for a continuous function {{mvar|g}}, if the sequence {{math|{''X<sub>n</sub>''} }} converges in distribution to {{mvar|X}}, then {{math|{''g''(''X<sub>n</sub>'')} }} converges in distribution to {{math|''g''(''X'')}}.
** Note however that convergence in distribution of {{math|{''X<sub>n</sub>''} }} to {{mvar|X}} and {{math|{''Y<sub>n</sub>''} }} to {{mvar|Y}} does in general ''not'' imply convergence in distribution of {{math|{''X<sub>n</sub>'' + ''Y<sub>n</sub>''} }} to {{math|''X'' + ''Y''}} or of {{math|{''X<sub>n</sub>Y<sub>n</sub>''} }} to {{mvar|XY}}.
* [[Lévy’s continuity theorem]]: The sequence {{math|{''X<sub>n</sub>''} }} converges in distribution to {{mvar|X}} if and only if the sequence of corresponding [[characteristic function (probability theory)|characteristic function]]s {{math|{''φ<sub>n</sub>''} }} [[pointwise convergence|converges pointwise]] to the characteristic function {{mvar|φ}} of {{mvar|X}}.
* Convergence in distribution is [[metrizable]] by the [[Lévy–Prokhorov metric]].
* A natural link to convergence in distribution is the [[Skorokhod's representation theorem]].