Convergence in distribution
A seq. X₁, X₂, ... of R-valued random vars. converges in distribution (or converges weakly, or in law) to a random var. X
⇔ lim Fₙ(x) = F(x) as n→∞ for all continuity points x of F
⇔ E(f(Xₙ))→E(f(X)) for all bounded Lipschitz f
⇔ liminf E(f(Xₙ))≥E(f(X)) for all non-negative continuous f
⇔ liminf P(Xₙ∈G)≥P(X∈G) for every open set G
⇔ limsup P(Xₙ∈F)≤P(X∈F) for every closed set F
⇔ P(Xₙ∈B)→P(X∈B) for all continuity sets B of X (i.e., P(X∈∂B)=0)
How many can you prove?
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