From Wikipedia

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

  • $E_nf → Ef$ for all bounded, continuous functions $f$;
  • $E_nf → Ef$ for all bounded and Lipschitz functions $f$;
  • $\limsup E_nf ≤ Ef$ for every upper semi-continuous function $f$ bounded from above;
  • $\liminf E_nf ≥ Ef$ for every lower semi-continuous function $f$ 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 $P$.

Did explained why the convergence mode is called weak*:

In the page you quote, Wikipedia defines weak convergence of (probability) measures. This mode of convergence should be called weak* rather than weak because it refers to the convergence against any bounded continuous functions, and the space $M_1$ of probability measures is included in the dual of the space $C_b$ of bounded continuous functions. That is, $(\mu_n)$ in $M_1$ converges to $\mu$ in $M_1$ if and only if $\int f\mathrm d\mu_n\to\int f\mathrm d\mu$ for every $f$ in $C_b$ and this is a weak* convergence because $M_1\subset(C_b)^*$.

Now I wonder why this convergence mode is called weak? Can "weak" be explained from the point of view of functional analysis, or is it explained from some other point of view?

Thanks and regards!

  • 1
    $\begingroup$ If $P_n$ is the distribution of a random variable $X_n$, then this weak form of convergence is maybe called weak because it's the weakest form of convergence of random variables (that I know of), i.e. it is implied by both convergence almost surely (which is often called strong convergence) and convergence in probability. This is the only explanation I can give at least. $\endgroup$ – Stefan Hansen Feb 23 '13 at 9:46
  • $\begingroup$ @StefanHansen: Thanks! That makes sense in some sense. $\endgroup$ – Tim Feb 23 '13 at 13:58

I have taken both real analysis and probability theory. So this question has bugged me for a while. This is just my 2 cents.

  • The weak* convergence point of view comes fromt the Riesz Representation Theorem: If X is a locally compact Hausdorff space, $M(X)$ is the space of complex Borel measures on X with the total variation norm, and C_0(X) is the space of functions that vanish at infinity on X. Then M(X) is isometrically isomorphic to the dual space of $C_0(X)$. Then the theory of normed vector spaces gives you weak* topology on M(X). So in Probability theory, if X is LCH, then convergence in distribution coincides with weak* convergence.

Reference: Chapter 7 of Folland Real Analysis: Modern Techniques and Their Applications.

  • The weak convergence point of view comes directly from the weak topology generated by bounded continuous functions. If X is a topological space, and P(X) is the space of probability measure on X. Then $BC(X)$ (bounded continuous functions on $X$) generates a weak topology on P(X). Convergence in the weak topology is just convergence in distribution. So the name. Even in this generality, you have a portion of the Portmanteau Theorem. There are text books that require X to be metric space, then the theory expands. Many textbooks require X to be polish, then this weak topology on P(X) is metrizable. There is no explicit assumption of LCH, and there is no Riesz Representation Theorem to give you a weak* topology.

Reference: Wikipedia, other websites, google books, and online lecture notes.

Best, Xiang

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Both topologies are defined on a (subset of a) dual space via pairing with the pre-dual. The difference is "algebraic" vs. functional-analytic pairing used in defining the respective topologies.

Let $X$ be a metric space. Consider the pairing used to define the weak convergence of Borel probability measures. The space of probability measures on $X$ is identified as a subset of $C_b(X)^*_+$, the positive cone of the vector space dual of $C_b(X)$. (It's a proper subset in general unless $X$ is compact---e.g. $X=\mathbb{R}$, $f \mapsto \lim \sup f$ is an element of the algebraic dual but not given by a measure.)

For a LCH $X$, weak-* convergence of Radon (i.e. Borel regular) probability measures is given by the pairing with the Banach space $C_c(X)$. Riesz representation characterizes the Radon probability measures as $C_c(X)^*_+ \, \cap\, C_b(X)^*_1$, the positive cone in the unit ball of $C_c(X)^*_+$.

The LCH property and the additional functional analytic structure imposes stronger regularity property on the family of measures considered---inner approximation by compact sets on a LCH space rather than merely closed sets on a metric space.

On the more concrete level, the the pairing with $C_c(X)$ cannot tell what happens at $\infty$. Take a sequence $x_n \rightarrow \infty$ on the real line. Then $x_n \rightarrow 0$ weak-* but not weakly. Indeed, weak topology coincides with the metric topology on $X$ when restricted to point-masses.

In probability applications, one typical does not have the LCH property---e.g. $X$ is the space of cadlag functions on $[0,1]$.

The reason that one restricts to $C_c(X)$ in the functional analytic context is that the Banach space dual of $C_b(X)$ is somewhat funny. E.g. when $X$ is normal, $C_b(X)^*$ can be identified with the space of finitely additive regular bounded Borel measures on $X$.

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