Proofs of convergence of random variables

This article is supplemental for “<a href="/facts/Convergence_of_random_variables/pWdatFlY">Convergence of random variables</a>” and provides proofs for selected results.
Several results will be established using the <a href="/facts/Portmanteau_lemma/fb0I0KAL">portmanteau lemma</a>: A sequence {Xn} converges in distribution to X if and only if any of the following conditions are met:

<ol>
 <li> 
 
 
 
 
 E
 
 [
 f
 (
 
 X
 
 n
 
 
 )
 ]
 →
 
 E
 
 [
 f
 (
 X
 )
 ]
 
 
 {\displaystyle \mathbb {E} [f(X_{n})]\to \mathbb {E} [f(X)]}
 
 for all <a href="/facts/Bounded_function/whVjbtuk">bounded</a>, <a href="/facts/Continuous_function/sKbl02pB">continuous functions</a> 
 
 
 
 f
 
 
 {\displaystyle f}
 
;
 </li><li> 
 
 
 
 
 E
 
 [
 f
 (
 
 X
 
 n
 
 
 )
 ]
 →
 
 E
 
 [
 f
 (
 X
 )
 ]
 
 
 {\displaystyle \mathbb {E} [f(X_{n})]\to \mathbb {E} [f(X)]}
 
 for all bounded, <a href="/facts/Lipschitz_function/Hw20EPEU">Lipschitz functions</a> 
 
 
 
 f
 
 
 {\displaystyle f}
 
;
 </li><li> 
 
 
 
 lim sup
 Pr
 ⁡
 (
 
 X
 
 n
 
 
 ∈
 C
 )
 ≤
 Pr
 ⁡
 (
 X
 ∈
 C
 )
 
 
 {\displaystyle \limsup \operatorname {Pr} (X_{n}\in C)\leq \operatorname {Pr} (X\in C)}
 
 for all <a href="/facts/Closed_set/upYnRTUj">closed sets</a> 
 
 
 
 C
 
 
 {\displaystyle C}
 
;
</li></ol>

Proofs of convergence of random variables open-in-new

Proofs of convergence of random variables