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Equivalence of measures

Convergence of measures

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

Let (X, mathcal{F}) be a measurable space. If the collection of all measures (or, frequently, just probability measures) on (X, mathcal{F}) can be given some kind of metric, then convergence in this metric is usually referred to as strong convergence. Examples include the Radon metric
rho (mu, nu) := sup left{ left. int_{X} f(x) , mathrm{d} (mu - nu) (x) right| mathrm{continuous,} f : X to [-1, 1] subset mathbb{R} right}
and the total variation metric
tau (mu, nu) := sup left{ left. | mu (A) - nu (A) | right| A in mathcal{F} right}.

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 (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 μP(Ω). Then the following conditions are all equivalent:

  1. lim_{n to infty} int_{Omega} f , mathrm{d} mu_{n} = int_{Omega} f , mathrm{d} mu for all bounded and continuous functions f : Ω → R (sometimes referred to as "test functions");
  2. limsupn→∞ μn(C) ≤ μ(C) for all closed subsets C of Ω;
  3. liminfn→∞ μn(U) ≥ μ(U) for all open subsets U of Ω;
  4. 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;
  5. 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 xR at which F is continuous.

Definition and notation

If any (and hence all) of the above conditions hold, the sequence of measures (mu_{n})_{n = 1}^{infty} is said to converge weakly to mu. 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 mu_{n} Rightarrow mu, mu_{n} rightharpoonup mu and mu_{n} xrightarrow{mathcal{D}} mu..

Weak convergence of random variables

If (Omega, mathcal{F}, mathbb{P}) is a probability space and X_{n}, X : Omega to mathbb{X} are random variables, X_{n} is said to converge weakly (or in distribution or in law) to X as n to infty if the sequence of pushforward measures (X_{n})_{*} (mathbb{P}) converges weakly to X_{*} (mathbb{P}) in the sense of weak convergence of measures on mathbb{X}, as defined above.

References

  • 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.

See also

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