Editing Convergence of random variables (section)

=== Definition ===
A sequence {''X''<sub>''n''</sub>} of random variables '''converges in probability''' towards the random variable ''X'' if for all ''ε'' > 0

: <math>\lim_{n\to\infty}\mathbb{P}\big(|X_n-X| > \varepsilon\big) = 0.</math>

More explicitly, let ''P''<sub>''n''</sub>(''ε'') be the probability that ''X''<sub>''n''</sub> is outside the ball of radius ''ε'' centered at&nbsp;''X''. Then {{mvar|X<sub>n</sub>}} is said to converge in probability to ''X'' if for any {{math|''ε'' > 0}} and any ''δ''&nbsp;>&nbsp;0 there exists a number ''N'' (which may depend on ''ε'' and ''δ'') such that for all ''n''&nbsp;≥&nbsp;''N'', ''P''<sub>''n''</sub>(''ε'')&nbsp;<&nbsp;''δ'' (the definition of limit).

Notice that for the condition to be satisfied, it is not possible that for each ''n'' the random variables ''X'' and ''X''<sub>''n''</sub> are independent (and thus convergence in probability is a condition on the joint cdf's, as opposed to convergence in distribution, which is a condition on the individual cdf's), unless ''X'' is deterministic like for the weak law of large numbers. At the same time, the case of a deterministic ''X'' cannot, whenever the deterministic value is a discontinuity point (not isolated), be handled by convergence in distribution, where discontinuity points have to be explicitly excluded.

Convergence in probability is denoted by adding the letter ''p'' over an arrow indicating convergence, or using the "plim" probability limit operator:
{{NumBlk|:| <math>X_n \ \xrightarrow{p}\ X,\ \ X_n \ \xrightarrow{P}\ X,\ \ \underset{n\to\infty}{\operatorname{plim}}\, X_n = X.</math>|{{EquationRef|2}}}}

For random elements {''X''<sub>''n''</sub>} on a [[separable metric space]] {{math|(''S'', ''d'')}}, convergence in probability is defined similarly by<ref>{{harvnb|Dudley|2002|loc=Chapter 9.2, page 287}}</ref>
: <math>\forall\varepsilon>0, \mathbb{P}\big(d(X_n,X)\geq\varepsilon\big) \to 0.</math>