Convergence of measures 1

Convergence of measuresIn mathematics, more specifically measure theory, there are various notions of the convergence of measures. For anintuitive general sense of what is meant by convergence in measure, consider a sequence of measures n on a space,sharing a common collection of measurable sets. Such a sequence might represent an attempt to construct 'better andbetter' approximations to a desired measure that is difficult to obtain directly. The meaning of 'better and better' issubject to all the usual caveats for taking limits; for any error tolerance > 0 we require there be N sufficiently largefor n N to ensure the 'difference' between n and is smaller than . Various notions of convergence specifyprecisely 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 descriptionsThis section attempts to provide a rough intuitive description of three notions of convergence, using terminologydeveloped in calculus courses; this section is necessarily imprecise as well as inexact, and the reader should refer tothe formal clarifications in subsequent sections. In particular, the descriptions here do not address the possibility thatn(A) or (A) could be infinite or zero.The various notions of weak convergence formalize the assertion that the 'average value' of each 'nice' functionshould converge:

To formalize this requires a careful specification of the set of functions under consideration, the domain ofintegration A, and of what is meant by average value (see expectation). This notion treats convergence for differentfunctions f independently of one another, i.e. different functions f may require different values of N n to beapproximated equally well (thus, a form of non-uniform convergence).The notion of strong convergence formalizes the assertion that the measure of each measurable set should converge:

Intuitively, considering integrals of 'nice' functions, this notion provides more uniformity than weak convergence: forany > 0 and any upper bound M > 0, we can find an N so that n N ensures

for all integrable functions f with |f| bounded by M. (The definition of strong convergence does not require one tospecify a set of measurable functions, so the interpretation given here might not apply.) This notion of convergencestill allows N to vary with the set A.The notion of total variation convergence formalizes the assertion that the measure of all measurable sets shouldconverge uniformly, i.e. we do not allow the N described above to depend upon which set A we are measuring. (Thisis only a very rough description of total variation convergence, as additional technical care is necessary; see below.)

Convergence of measures 2

Total variation convergence of measuresThis is the strongest notion of convergence shown on this page and is defined as follows. Let be ameasurable space. The total variation distance between two (positive) measures and is then given by

Here the supremum is taken over f ranging over the set of all measurable functions from X to [1, 1]. This is incontrast, for example, to the Wasserstein metric, where the definition is of the same form, but the supremum is takenover f ranging over the set of measurable functions from X to [1, 1] which have Lipschitz constant at most 1; andalso in contrast to the Radon metric, where the supremum is taken over f ranging over the set of continuous functionsfrom X to [1, 1]. In the case where X 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 alwaysbetween 0 and 2.To illustrate the meaning of the total variation distance, consider the following thought experiment. Assume that weare given two probability measures and , as well as a random variable X. We know that X has law either or butwe do not know which one of the two. Assume that these two measures have prior probabilities 0.5 each of being thetrue law of X. Assume now that we are given one single sample distributed according to the law of X and that we arethen 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 n of measures defined on the same measure spaceis said to converge to a measure in total variation distance if for every > 0, there exists an N such that for all n >N, one has that

.

Strong convergence of measuresFor a measurable space, a sequence n is said to converge strongly to a limit if

for every set .For example, as a consequence of the RiemannLebesgue lemma, the sequence n of measures on the interval [1,1] given by n(dx) = (1+ sin(nx))dx converges strongly to Lebesgue measure, but it does not converge in totalvariation.

Convergence of measures 3

Weak convergence of measuresIn 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 ofmany types of convergence relating to the convergence of measures. It depends on a topology on the underlyingspace 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 portmanteautheorem.[1]

Definition. Let S be a metric space with its Borel -algebra . We say that a sequence of probabilitymeasures Pn on (S, ), n = 1, 2, ..., converges weakly to the probability measure P, and write

if any of the following equivalent conditions is true (here En denotes expectation with respect to Pnwhile E denotes expectation with respect to P):

Enf Ef for all bounded, continuous functions f; Enf Ef for all bounded and Lipschitz functions f; limsup Enf Ef for every upper semi-continuous function f bounded from above; liminf Enf Ef for every lower semi-continuous function f bounded from below; limsup Pn(C) P(C) for all closed sets C of space S; liminf Pn(U) P(U) for all open sets U of space S; lim Pn(A) = P(A) for all continuity sets A of measure P.

In the case S = R with its usual topology, if Fn, F denote the cumulative distribution functions of the measures Pn, Prespectively, then Pn converges weakly to P if and only if limn Fn(x) = F(x) for all points x R at which F iscontinuous.For example, the sequence where Pn is the Dirac measure located at 1/n converges weakly to the Dirac measurelocated at 0 (if we view these as measures on R with the usual topology), but it does not converge strongly. This isintuitively clear: we only know that 1/n is "close" to 0 because of the topology of R.This definition of weak convergence can be extended for S any metrizable topological space. It also defines a weaktopology on P(S), the set of all probability measures defined on (S,). The weak topology is generated by thefollowing basis of open sets:

where

If S is also separable, then P(S) is metrizable and separable, for example by the LvyProkhorov metric, if S is alsocompact or Polish, so is P(S).If S is separable, it naturally embeds into 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 , and .

Convergence of measures 4

Weak convergence of random variables

Let be a probability space and X be a metric space. If Xn, X: X is a sequence of random variablesthen Xn is said to converge weakly (or in distribution or in law) to X as n if the sequence of pushforwardmeasures (Xn)(P) converges weakly to X(P) in the sense of weak convergence of measures on X, as defined above.

References[1] Achim Klenke, Probability theory (2006) Springer-Verlag, ISBN 978-1-848000-047-6 doi:10.1007/978-1-848000-048-3

Ambrosio, L., Gigli, N. & Savar, G. (2005). Gradient Flows in Metric Spaces and in the Space of ProbabilityMeasures. Basel: ETH Zrich, Birkhuser Verlag. ISBN3-7643-2428-7.

Billingsley, Patrick (1995). Probability and Measure. New York, NY: John Wiley & Sons, Inc.ISBN0-471-00710-2.

Billingsley, Patrick (1999). Convergence of Probability Measures. New York, NY: John Wiley & Sons, Inc.ISBN0-471-19745-9.

Article Sources and Contributors 5

Article Sources and ContributorsConvergence of measures Source: http://en.wikipedia.org/w/index.php?oldid=577518765 Contributors: Derveish, Diegotorquemada, Hairer, Igny, J04n, Linas, Melcombe, Michael Hardy, PaulAugust, Piyush Sriva, Radagast83, Stpasha, Sullivan.t.j, Thenub314, 37 anonymous edits

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Convergence of measuresInformal descriptionsTotal variation convergence of measuresStrong convergence of measuresWeak convergence of measures Weak convergence of random variables

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