Lévy–Prokhorov metric

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In mathematics, the Lévy–Prokhorov metric (sometimes known just as the Prokhorov metric) is a metric (i.e., a definition of distance) on the collection of probability measures on a given metric space. It is named after the French mathematician Paul Lévy and the Soviet mathematician Yuri Vasilyevich Prokhorov; Prokhorov introduced it in 1956 as a generalization of the earlier Lévy metric.

Definition

Let (M,d) be a metric space with its Borel sigma algebra (M). Let 𝒫(M) denote the collection of all probability measures on the measurable space (M,(M)).

For a subset AM, define the ε-neighborhood of A by

Aε:={pM|qA, d(p,q)<ε}=pABε(p).

where Bε(p) is the open ball of radius ε centered at p.

The Lévy–Prokhorov metric π:𝒫(M)2[0,+) is defined by setting the distance between two probability measures μ and ν to be

π(μ,ν):=inf{ε>0|μ(A)ν(Aε)+ε and ν(A)μ(Aε)+ε for all A(M)}.

For probability measures clearly π(μ,ν)1.

Some authors omit one of the two inequalities or choose only open or closed A; either inequality implies the other, and (A¯)ε=Aε, but restricting to open sets may change the metric so defined (if M is not Polish).

Properties

  • If (M,d) is separable, convergence of measures in the Lévy–Prokhorov metric is equivalent to weak convergence of measures. Thus, π is a metrization of the topology of weak convergence on 𝒫(M).
  • The metric space (𝒫(M),π) is separable if and only if (M,d) is separable.
  • If (𝒫(M),π) is complete then (M,d) is complete. If all the measures in 𝒫(M) have separable support, then the converse implication also holds: if (M,d) is complete then (𝒫(M),π) is complete. In particular, this is the case if (M,d) is separable.
  • If (M,d) is separable and complete, a subset 𝒦𝒫(M) is relatively compact if and only if its π-closure is π-compact.
  • If (M,d) is separable, then π(μ,ν)=inf{α(X,Y):Law(X)=μ,Law(Y)=ν}, where α(X,Y)=inf{ε>0:(d(X,Y)>ε)ε} is the Ky Fan metric.[1][2]

Relation to other distances

Let (M,d) be separable. Then

See also

Notes

  1. Dudley 1989, p. 322
  2. Račev 1991, p. 159
  3. Gibbs, Alison L.; Su, Francis Edward: On Choosing and Bounding Probability Metrics, International Statistical Review / Revue Internationale de Statistique, Vol 70 (3), pp. 419-435, Lecture Notes in Math., 2002.
  4. Račev 1991, p. 175

References