In mathematics, there are various notions of the convergence of measures in measure theory. Broadly speaking, there are two kinds of convergence, strong convergence and weak convergence.
Strong convergence of measures
be a measurable space
. If the collection of all measures (or, frequently, just probability measures
can be given some kind of metric
, then convergence
in this metric is usually referred to as strong convergence
. Examples include the Radon metric
and the total variation
Weak convergence of measures
, 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 (at least) five definitions of weak convergence of a sequence of measures, some of which are more general than others. The following equivalence result is sometimes known as the portmanteau theorem, and shows the equivalence of four such definitions for probability measures on a general metrizable topological space, and a fifth condition, which makes sense only for distributions on the real line.
Let (Ω, T) be a topological space with its Borel σ-algebra Borel(Ω), and let P(Ω) denote the collection of all probability measures defined on (Ω, Borel(Ω)). Consider here the case of metrizable Ω (we need some restriction of the topology and second countable is not sufficient). If Ω is also separable, P(Ω) (with the weak topology defined below) is metrizable, for example by the Lévy-Prokhorov metric. Let μn, n = 1, 2, ..., be a sequence in P(Ω) and let μ ∈ P(Ω). Then the following conditions are all equivalent:
- for all bounded and continuous functions f : Ω → R (sometimes referred to as "test functions");
- limsupn→∞ μn(C) ≤ μ(C) for all closed subsets C of Ω;
- liminfn→∞ μn(U) ≥ μ(U) for all open subsets U of Ω;
- limn→∞ μn(A) = μ(A) for all so-called "μ-continuity" subsets A of Ω: those sets A with μ(∂A) = 0, where ∂A denotes the boundary of A;
- in the case Ω = R with its usual topology, if Fn, F denote the cumulative distribution functions of the measures μn, μ respectively, then limn→∞ Fn(x) = F(x) for all points x ∈ R at which F is continuous.
Definition and notation
If any (and hence all) of the above conditions hold, the sequence of measures
is said to converge weakly
. Weak convergence is also known as narrow convergence
, convergence in distribution
and convergence in law
(the terms "convergence in distribution/law" are more frequently used when discussing weak convergence of random variables, as in the next section).
There are many "arrow notations" for this kind of convergence: the most frequently used are , and .
Weak convergence of random variables
is a probability space
are random variables
is said to converge weakly
(or in distribution
or in law
if the sequence of pushforward measures
converges weakly to
in the sense of weak convergence of measures on
, as defined above.
- Ambrosio, L., Gigli, N. & Savaré, G. (2005). Gradient Flows in Metric Spaces and in the Space of Probability Measures. Basel: ETH Zürich, Birkhäuser Verlag. ISBN 3-7643-2428-7.
- Billingsley, Patrick (1995). Probability and Measure. New York, NY: John Wiley & Sons, Inc.. ISBN 0-471-00710-2.
- Billingsley, Patrick (1999). Convergence of Probability Measures. New York, NY: John Wiley & Sons, Inc.. ISBN 0-471-19745-9.