Editing Convergence of random variables (section)

===Definition===
A sequence <math>X_1, X_2, \ldots </math> of real-valued [[random variable]]s, with [[cumulative distribution function]]s <math>F_1, F_2, \ldots </math>, is said to '''converge in distribution''', or '''converge weakly''', or '''converge in law''' to a random variable {{mvar|X}} with [[cumulative distribution function]] {{mvar|F}} if

: <math>\lim_{n\to\infty} F_n(x) = F(x),</math>

for every number <math>x \in \mathbb{R}</math> at which <math> F </math> is [[continuous function|continuous]].

The requirement that only the continuity points of <math> F </math> should be considered is essential. For example, if <math> X_n </math> are distributed [[Uniform distribution (continuous)|uniformly]] on intervals <math> \left( 0,\frac{1}{n} \right)  </math>, then this sequence converges in distribution to the [[degenerate distribution|degenerate]] random variable <math>  X=0 </math>. Indeed, <math>  F_n(x) =  0 </math> [[existential quantification|for all]] <math> n  </math> when <math>   x\leq 0</math>, and <math> F_n(x) = 1  </math> for all <math> x \geq \frac{1}{n}  </math>when <math> n > 0  </math>. However, for this limiting random variable <math> F(0) = 1  </math>, even though <math> F_n(0) = 0  </math> for all <math> n </math>. Thus the convergence of cdfs fails at the point <math> x=0  </math> where <math> F  </math> is discontinuous.

Convergence in distribution may be denoted as

{{NumBlk|:|<math>\begin{align}
  {}
  & X_n \ \xrightarrow{d}\ X,\ \ 
    X_n \ \xrightarrow{\mathcal{D}}\ X,\ \ 
    X_n \ \xrightarrow{\mathcal{L}}\ X,\ \ 
    X_n \ \xrightarrow{d}\ \mathcal{L}_X, \\
  & X_n \rightsquigarrow X,\ \ 
    X_n \Rightarrow X,\ \ 
    \mathcal{L}(X_n)\to\mathcal{L}(X),\\ 
  \end{align}</math>
|{{EquationRef|1}}}}

where <math>\scriptstyle\mathcal{L}_X</math> is the law (probability distribution) of {{mvar|X}}. For example, if {{mvar|X}} is standard normal we can write <math style="height:1.5em;position:relative;top:-.3em">X_n\,\xrightarrow{d}\,\mathcal{N}(0,\,1)</math>.

For [[random vector]]s <math>\left\{ X_1,X_2,\dots \right\}\subset \mathbb{R}^k</math> the convergence in distribution is defined similarly. We say that this sequence '''converges in distribution''' to a random {{mvar|k}}-vector {{mvar|X}} if
: <math>\lim_{n\to\infty} \mathbb{P}(X_n\in A) = \mathbb{P}(X\in A)</math>
for every <math>A\subset \mathbb{R}^k</math> which is a [[continuity set]] of {{mvar|X}}.

The definition of convergence in distribution may be extended from random vectors to more general [[random element]]s in arbitrary [[metric space]]s, and even to the “random variables” which are not measurable — a situation which occurs for example in the study of [[empirical process]]es. This is the “weak convergence of laws without laws being defined” — except asymptotically.<ref>{{harvnb|Bickel|Klaassen|Ritov|Wellner|1998|loc=A.8, page 475}}</ref>

In this case the term '''weak convergence''' is preferable (see [[weak convergence of measures]]), and we say that a sequence of random elements {{math|{''X<sub>n</sub>''} }} converges weakly to {{mvar|X}} (denoted as {{math|''X<sub>n</sub>'' ⇒ ''X''}}) if
: <math>\mathbb{E}^*h(X_n) \to \mathbb{E}\,h(X)</math>
for all continuous bounded functions {{mvar|h}}.<ref>{{harvnb|van der Vaart|Wellner|1996|page=4}}</ref> Here E* denotes the ''outer expectation'', that is the expectation of a “smallest measurable function {{mvar|g}} that dominates {{math|''h''(''X<sub>n</sub>'')}}”.