weak convergence for probability measures;
1.Helly's selection theorem: Let A be an infinite collection of sub-prob measures on （R,B(R)）. Then there exist a sequence
{ μ_n } ⊂ A and a sub-prob measure μ such that μ_n → μ vaguely.
2. Let { μ_n } (n>=1) be a sequence of prob measures on （R,B(R)）. Then μ_n → μ weakly iff { μ_n } (n>=1) is tight and all weakly convergent subsequence of { μ_n } (n>=1) converge to the same limiting prob measure μ.
Firstly, the notion of tightness of prob measures or r.v.s is analogous to the notion of boundedness of a sequence of real numbers. For a sequence of r.v.s
{ X_n } (n>=1) , the condition of tightness requires that given ε >0 arbitrarily small, there exists an M=M_ ε such that for each n, X_n lies in [-M, M] with probability at least 1- ε . Thus, for a tight sequence of r.v.s, no positive mass can escape to +∞ or -∞.
Secondly, like a bounded sequence of real numbers , a tight sequence of r.v.s may not converge weakly, but has one or more convergent subsequences.
Weak convergence on Polish space:
1. Portmanteau's theorem (characterization of weak convergence)
2. Prohorov-Varadarajan theorem
(characterization of tightness)
Skorohod's theorem:
If μ_n → μ weakly, then there exist r.v.s X_n, n>=1,and X, such that X_n has distribution of μ_n , X has distribution of μ, and X_n→X with probability 1.
The continuous mapping theorem.