Weak convergence of measures

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In mathematics and statistics, weak convergence (also known as narrow convergence) is one of many types of convergence relating to the convergence of measures.

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[edit] Equivalent definitions: the portmanteau theorem

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 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)). Let μn, n = 1, 2, ..., be a sequence in P(Ω) and let μ ∈ P(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 x ∈ R at which F is continuous.

[edit] 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 μ. 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} \, \begin{matrix} {\,}_\mathcal{D} \\ {\,}^{\longrightarrow} \\ \quad \end{matrix} \, \mu..

[edit] 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, Xn 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.

[edit] 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. 

[edit] See also

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