Convergence of Measures - Weak Convergence of Measures

Weak Convergence of Measures

In mathematics and statistics, weak convergence (also known as narrow convergence or weak-* convergence, which is a more appropriate name from the point of view of functional analysis, but less frequently used) is one of many types of convergence relating to the convergence of measures. It depends on a topology on the underlying space and thus is not a purely measure theoretic notion.

There are several equivalent definitions of weak convergence of a sequence of measures, some of which are (apparently) more general than others. The equivalence of these conditions is sometimes known as the portmanteau theorem.

Definition. Let S be a metric space with its Borel σ-algebra Σ. We say that a sequence of probability measures P_n on (S, Σ), n = 1, 2, ..., converges weakly to the probability measure P, and write

if any of the following equivalent conditions is true (here E_n denotes expectation with respect to while E denotes expectation with respect to ):

• E_nƒ → Eƒ for all bounded, continuous functions ƒ;
• E_nƒ → Eƒ for all bounded and Lipschitz functions ƒ;
• limsup E_nƒ ≤ Eƒ for every upper semi-continuous function ƒ bounded from above;
• liminf E_nƒ ≥ Eƒ for every lower semi-continuous function ƒ bounded from below;
• limsup P_n(C) ≤ P(C) for all closed sets C of space S;
• liminf P_n(U) ≥ P(U) for all open sets U of space S;
• lim P_n(A) = P(A) for all continuity sets A of measure P.

In the case S = R with its usual topology, if F_n, F denote the cumulative distribution functions of the measures P_n, P respectively, then P_n converges weakly to P if and only if lim_n→∞ F_n(x) = F(x) for all points x ∈ R at which F is continuous.

For example, the sequence where P_n is the Dirac measure located at 1/n converges weakly to the Dirac measure located at 0 (if we view these as measures on R with the usual topology), but it does not converge strongly. This is intuitively clear: we only know that 1/n is "close" to 0 because of the topology of R.

This definition of weak convergence can be extended for S any metrizable topological space. It also defines a weak topology on P(S), the set of all probability measures defined on (S, Σ). The weak topology is generated by the following basis of open sets:

where

If S is also separable, then P(S) is metrizable and separable, for example by the Lévy–Prokhorov metric, if S is also compact or Polish, so is P(S).

If S is separable, it naturally embeds into P(S) as the (closed) set of dirac measures, and its convex hull is dense.

There are many "arrow notations" for this kind of convergence: the most frequently used are, and .