Convergence of measures
In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by convergence of measures, consider a sequence of measures on a space, sharing a common collection of measurable sets. Such a sequence might represent an attempt to construct 'better and better' approximations to a desired measure that is difficult to obtain directly. The meaning of 'better and better' is subject to all the usual caveats for taking limits; for any error tolerance we require there be sufficiently large for to ensure the 'difference' between and is smaller than. Various notions of convergence specify precisely what the word 'difference' should mean in that description; these notions are not equivalent to one another, and vary in strength.
Three of the most common notions of convergence are described below.
Informal descriptions
This section attempts to provide a rough intuitive description of three notions of convergence, using terminology developed in calculus courses; this section is necessarily imprecise as well as inexact, and the reader should refer to the formal clarifications in subsequent sections. In particular, the descriptions here do not address the possibility that the measure of some sets could be infinite, or that the underlying space could exhibit pathological behavior, and additional technical assumptions are needed for some of the statements. The statements in this section are however all correct if is a sequence of probability measures on a Polish space.The various notions of convergence formalize the assertion that the 'average value' of each 'sufficiently nice' function should converge:
To formalize this requires a careful specification of the set of functions under consideration and how uniform the convergence should be.
The notion of weak convergence requires this convergence to take place for every continuous bounded function.
This notion treats convergence for different functions independently of one another, i.e., different functions may require different values of to be approximated equally well.
The notion of setwise convergence formalizes the assertion that the measure of each measurable set should converge:
Again, no uniformity over the set is required.
Intuitively, considering integrals of 'nice' functions, this notion provides more uniformity than weak convergence. As a matter of fact, when considering sequences of measures with uniformly bounded
variation on a Polish space, setwise convergence implies the convergence for any bounded measurable function.
As before, this convergence is non-uniform in.
The notion of total variation convergence formalizes the assertion that the measure of all measurable sets should converge uniformly, i.e. for every there exists such that for every and for every measurable set. As before, this implies convergence of integrals against bounded measurable functions, but this time convergence is uniform over all functions bounded by any fixed constant.
Total variation convergence of measures
This is the strongest notion of convergence shown on this page and is defined as follows. Let be a measurable space. The total variation distance between two measures and is then given byHere the supremum is taken over ranging over the set of all measurable functions from to. This is in contrast, for example, to the Wasserstein metric, where the definition is of the same form, but the supremum is taken over ranging over the set of those measurable functions from to which have Lipschitz constant at most 1; and also in contrast to the Radon metric, where the supremum is taken over ranging over the set of continuous functions from to. In the case where is a Polish space, the total variation metric coincides with the Radon metric.
If and are both probability measures, then the total variation distance is also given by
The equivalence between these two definitions can be seen as a particular case of the Monge–Kantorovich duality. From the two definitions above, it is clear that the total variation distance between probability measures is always between 0 and 2.
To illustrate the meaning of the total variation distance, consider the following thought experiment. Assume that we are given two probability measures and, as well as a random variable. We know that has law either or but we do not know which one of the two. Assume that these two measures have prior probabilities 0.5 each of being the true law of. Assume now that we are given one single sample distributed according to the law of and that we are then asked to guess which one of the two distributions describes that law. The quantity
then provides a sharp upper bound on the prior probability that our guess will be correct.
Given the above definition of total variation distance, a sequence of measures defined on the same measure space is said to converge to a measure in total variation distance if for every, there exists an such that for all, one has that
Setwise convergence of measures
For a measurable space, a sequence is said to converge setwise to a limit iffor every set.
Typical arrow notations are and.
For example, as a consequence of the Riemann–Lebesgue lemma, the sequence of measures on the interval given by converges setwise to Lebesgue measure, but it does not converge in total variation.
In a measure theoretical or probabilistic context setwise convergence is often referred to as strong convergence. This can lead to some ambiguity because in functional analysis, strong convergence usually refers to convergence with respect to a norm.
Weak convergence of measures
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 more general than others. The equivalence of these conditions is sometimes known as the Portmanteau theorem.
Definition. Let be a metric space with its Borel -algebra. A bounded sequence of positive probability measures on is said to converge weakly to a probability measure if any of the following equivalent conditions is true :
- for all bounded, continuous functions ;
- for all bounded and Lipschitz functions ;
- for every upper semi-continuous function bounded from above;
- for every lower semi-continuous function bounded from below;
- for all closed sets of space ;
- for all open sets of space ;
- for all continuity sets of measure.
For example, the sequence where is the Dirac measure located at converges weakly to the Dirac measure located at 0, but it does not converge setwise. This is intuitively clear: we only know that is "close" to because of the topology of.
This definition of weak convergence can be extended for any metrizable topological space. It also defines a weak topology on, the set of all probability measures defined on. The weak topology is generated by the following basis of open sets:
where
If is also separable, then is metrizable and separable, for example by the Lévy–Prokhorov metric. If is also compact or Polish, so is.
If is separable, it naturally embeds into as the 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,, and.
Weak convergence of random variables
Let be a probability space and X be a metric space. If is a sequence of random variables then Xn is said to converge weakly to the random variable X: Ω → X as if the sequence of pushforward measures ∗ converges weakly to X∗ in the sense of weak convergence of measures on X, as defined above.Comparison with vague convergence
Let be a metric space. The following spaces of test functions are commonly used in the convergence of probability measures.- the class of continuous functions each vanishing outside a compact set.
- the class of continuous functions such that
- the class of continuous bounded functions
Vague Convergence
A sequence of measures converges vaguely to a measure if for all, .Weak Convergence
A sequence of measures converges weakly to a measure if for all, .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 converges vaguely to a probability measure if and only if converges weakly to.
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 measures may not necessarily converge vaguely to a true probability measure, but rather to a sub-probability measure. Thus, a sequence of probability measures such that where 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 be a topological vector space or Banach space.
- A sequence in converges weakly to if as for all. One writes as.
- A sequence of converges in the weak-* topology to provided that for all. That is, convergence occurs in the point-wise sense. In this case, one writes as.