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 equivalent 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.

Let (Ω,T) be a topological space with its Borel sigma algebra σ(T), and let \mathcal{P}(\Omega) denote the collection of all probability measures defined on (Ω,σ(T)). Let (\mu_{n})_{n = 1}^{\infty} be a sequence in \mathcal{P}(\Omega) and let \mu \in \mathcal{P}(\Omega). 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 : \Omega \to \mathbb{R};
  2. \limsup_{n \to \infty} \mu_{n} (C) \leq \mu (C) for all closed sets C \subseteq \Omega;
  3. \liminf_{n \to \infty} \mu_{n} (U) \geq \mu (U) for all open sets U \subseteq \Omega;
  4. \lim_{n \to \infty} \mu_{n} (A) = \mu (A) for all "μ-continuity" sets A: sets A \subseteq \Omega with \mu (\partial A) = 0, where \partial A denotes the boundary of A;
  5. in the case \Omega = \mathbb{R}, if Fn,F denote the cumulative distribution functions of the measures μn respectively, then \lim_{n \to \infty} F_{n} (x) = F(x) for all points x \in \mathbb{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 push forward 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 3-764-32428-7.
  • Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc.. ISBN 0-471-00710-2.
  • Billingsley, Patrick (1999). Convergence of Probability Measures. New York: John Wiley & Sons, Inc.. ISBN 0-471-19745-9.

[edit] See also