Proofs of convergence of random variables

This article is supplemental for “Convergence of random variables” and provides proofs for selected results. Several results will be established using the portmanteau lemma: A sequence {Xn} converges in distribution to X if and only if any of the following conditions are met: E [ f ( X n ) ] → E [ f ( X ) ] {\displaystyle \mathbb {E} [f(X_{n})]\to \mathbb {E} [f(X)]} for all bounded, continuous functions f {\displaystyle f} ; E [ f ( X n ) ] → E [ f ( X ) ] {\displaystyle \mathbb {E} [f(X_{n})]\to \mathbb {E} [f(X)]} for all bounded, Lipschitz functions f {\displaystyle f} ; 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 closed sets C {\displaystyle C} ; == Convergence almost surely implies convergence in probability == X n → a s X ⇒ X n → p X {\displaystyle X_{n}\ {\overset {\mathrm {as} }{\rightarrow }}\ X\quad \Rightarrow \quad X_{n}\ {\overset {p}{\rightarrow }}\ X} Proof: If { X n } {\displaystyle \{X_{n}\}} converges to X {\displaystyle X} almost surely, it means that the set of points O = { ω : lim X n ( ω ) ≠ X ( ω ) } {\displaystyle O=\{\omega :\lim X_{n}(\omega )\neq X(\omega )\}} has measure zero.

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