In
mathematics and
statistics,
weak convergence 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 \sigma-algebra \Sigma. A bounded sequence of positive
probability measures P_n\, (n = 1, 2, \dots) on (S, \Sigma) is said to
converge weakly to a probability measure P (denoted P_n\Rightarrow P) if any of the following equivalent conditions is true (here \operatorname{E}_n denotes expectation or the integral with respect to P_n, while \operatorname{E} denotes expectation or the integral with respect to P): • \operatorname{E}_n[f] \to \operatorname{E}[f] for all
bounded,
continuous functions f; • \operatorname{E}_n[f] \to \operatorname{E}[f] for all bounded and
Lipschitz functions f; • \limsup \operatorname{E}_n[f] \le \operatorname{E}[f] for every
upper semi-continuous function f bounded from above; • \liminf \operatorname{E}_n[f] \ge \operatorname{E}[f] for every
lower semi-continuous function f bounded from below; • \limsup P_n(C) \le P(C) for all
closed sets C of space S; • \liminf P_n(U) \ge P(U) for all
open sets U of space S; • \lim P_n(A) = P(A) for all
continuity sets A of measure P. In the case S and \mathbb{R} (with its usual topology) are homeomorphic , if F_n and F denote the
cumulative distribution functions of the measures P_n and P, respectively, then P_n converges weakly to P if and only if \lim_{n \to \infty} F_n(x) = F(x) for all points x \in \mathbb{R} at which F is continuous. For example, the sequence where P_n is the
Dirac measure located at 1/n converges weakly to the Dirac measure located at 0 (if we view these as measures on \mathbb{R} with the usual topology), but it does not converge setwise. This is intuitively clear: we only know that 1/n is "close" to 0 because of the topology of \mathbb{R}. This definition of weak convergence can be extended for S any
metrizable topological space. It also defines a weak topology on \mathcal{P}(S), the set of all probability measures defined on (S,\Sigma). The weak topology is generated by the following basis of open sets: :\left\{ \ U_{\varphi, x, \delta} \ \left| \quad \varphi : S \to \mathbb{R} \text{ is bounded and continuous, } x \in \mathbb{R} \text{ and } \delta > 0 \ \right. \right\}, where :U_{\varphi, x, \delta} := \left\{ \ \mu \in \mathcal{P}(S) \ \left| \quad \left| \int_S \varphi \, \mathrm{d} \mu - x \right| If S is also
separable, then \mathcal{P}(S) is metrizable and separable, for example by the
Lévy–Prokhorov metric. If S is also compact or
Polish, so is \mathcal{P}(S). If S is separable, it naturally embeds into \mathcal{P}(S) as the (closed) set of
Dirac measures, and its
convex hull is
dense. There are many "arrow notations" for this kind of convergence: the most frequently used are P_{n} \Rightarrow P, P_{n} \rightharpoonup P, P_{n} \xrightarrow{w} P and P_{n} \xrightarrow{\mathcal{D}} P.
Weak convergence of random variables Let (\Omega, \mathcal{F}, \mathbb{P}) be a
probability space and
X be a metric space. If is a sequence of
random variables then
Xn is said to
converge weakly (or
in distribution or
in law) to the random variable
X: Ω →
X as if the sequence of
pushforward measures (
Xn)∗(
P) converges weakly to
X∗(
P) in the sense of weak convergence of measures on
X, as defined above.
Comparison with vague convergence Let X be a metric space (for example \mathbb{R} or [0,1]). The following spaces of test functions are commonly used in the convergence of probability measures. • C_c(X) the class of continuous functions f each vanishing outside a compact set. • C_0(X) the class of continuous functions f such that \lim _{|x| \rightarrow \infty} f(x)=0 • C_B(X) the class of continuous bounded functions We have C_c \subset C_0 \subset C_B \subset C. Moreover, C_0 is the closure of C_c with respect to uniform convergence.
Vague Convergence A sequence of measures \left(\mu_n\right)_{n \in \mathbb{N}}
converges vaguely to a measure \mu if for all f \in C_c(X), \int_X f \, d \mu_n \rightarrow \int_X f \, d \mu.
Weak Convergence A sequence of measures \left(\mu_n\right)_{n \in \mathbb{N}}
converges weakly to a measure \mu if for all f \in C_B(X), \int_X f \, d \mu_n \rightarrow \int_X f \, d \mu. In general, these two convergence notions are not equivalent. In a probability setting, vague convergence and weak convergence of probability measures are equivalent assuming
tightness. That is, a tight sequence of probability measures (\mu_n)_{n\in \mathbb{N}} converges
vaguely to a probability measure \mu if and only if (\mu_n)_{n \in \mathbb{N}} converges weakly to \mu. The weak limit of a sequence of probability measures, provided it exists, is a probability measure. In general, if tightness is not assumed, a sequence of probability (or sub-probability) measures may not necessarily converge
vaguely to a true probability measure, but rather to a sub-probability measure (a measure such that \mu(X)\leq 1). Thus, a sequence of probability measures (\mu_n)_{n\in \mathbb{N}} such that \mu_n \overset{v}{\to} \mu where \mu is not specified to be a probability measure is not guaranteed to imply weak convergence.
Weak convergence of measures as an example of weak-* convergence Despite having the same name as
weak convergence in the context of functional analysis, weak convergence of measures is actually an example of weak-* convergence. The definitions of weak and weak-* convergences used in functional analysis are as follows: Let V be a topological vector space or Banach space. • A sequence x_n in V
converges weakly to x if \varphi\left(x_n\right) \rightarrow \varphi(x) as n \to \infty for all \varphi \in V^*. One writes x_n \mathrel{\stackrel{w}{\rightarrow}} x as n \to \infty. • A sequence of \varphi_n \in V^*
converges in the weak-* topology to \varphi provided that \varphi_n(x) \rightarrow \varphi(x) for all x \in V. That is, convergence occurs in the point-wise sense. In this case, one writes \varphi_n \mathrel{\stackrel{w^*}{\rightarrow}} \varphi as n \to \infty. To illustrate how weak convergence of measures is an example of weak-* convergence, we give an example in terms of vague convergence (see above). Let X be a locally compact Hausdorff space. By the
Riesz-Representation theorem, the space M(X) of Radon measures is isomorphic to a subspace of the space of continuous linear functionals on C_0(X). Therefore, for each Radon measure \mu_n \in M(X), there is a linear functional \varphi_n \in C_0(X)^* such that \varphi_n(f)=\int_X f \, d \mu_n for all f \in C_0(X). Applying the definition of weak-* convergence in terms of linear functionals, the characterization of vague convergence of measures is obtained. For compact X , C_0(X)=C_B(X) , so in this case weak convergence of measures is a special case of weak-* convergence. ==See also==