Editing Convergence of random variables (section)

== Convergence in mean ==
Given a real number {{math|''r'' ≥ 1}}, we say that the sequence {{mvar|X<sub>n</sub>}} converges '''in the ''r''-th mean''' (or '''in the [[Lp space|''L<sup>r</sup>''-norm]]''') towards the random variable ''X'', if the {{mvar|r}}-th [[Moment (mathematics)|absolute moment]]s <math>\mathbb{E}</math>(|''X<sub>n</sub>''|<sup>''r ''</sup>) and <math>\mathbb{E}</math>(|''X''|<sup>''r ''</sup>) of {{mvar|X<sub>n</sub>}} and ''X'' exist, and
: <math>\lim_{n\to\infty} \mathbb{E}\left( |X_n-X|^r \right) = 0,</math>
where the operator E denotes the [[expected value]]. Convergence in {{mvar|r}}-th mean tells us that the expectation of the {{mvar|r}}-th power of the difference between <math>X_n</math> and <math>X</math> converges to zero.

This type of convergence is often denoted by adding the letter ''L<sup>r</sup>'' over an arrow indicating convergence:

{{NumBlk|:| <math>\overset{}{X_n \, \xrightarrow{L^r} \, X.}</math>|{{EquationRef|4}}}}

The most important cases of convergence in ''r''-th mean are:
* When {{mvar|X<sub>n</sub>}} converges in ''r''-th mean to ''X'' for ''r'' = 1, we say that {{mvar|X<sub>n</sub>}} converges '''in mean''' to ''X''.
* When {{mvar|X<sub>n</sub>}} converges in ''r''-th mean to ''X'' for ''r'' = 2, we say that {{mvar|X<sub>n</sub>}} converges '''in mean square''' (or '''in quadratic mean''') to ''X''.

Convergence in the ''r''-th mean, for ''r'' ≥ 1, implies convergence in probability (by [[Markov's inequality]]). Furthermore, if ''r'' > ''s'' ≥ 1, convergence in ''r''-th mean implies convergence in ''s''-th mean.  Hence, convergence in mean square implies convergence in mean.

Additionally, 
: <math>\overset{}{X_n \xrightarrow{L^r} X} \quad\Rightarrow\quad   \lim_{n \to \infty} \mathbb{E}[|X_n|^r] = \mathbb{E}[|X|^r]. </math>
The converse is not necessarily true, however it is true if <math>\overset{}{X_n \, \xrightarrow{p} \, X}</math> (by a more general version of [[Scheffé's lemma]]).