Lévy–Prokhorov metric explained

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 l{B} (M) . Let l{P} (M) denote the collection of all probability measures on the measurable space (M, l{B} (M)) .A \subseteq M , define the ε-neighborhood of A byA ^{\varepsilon} : = \{ p \in M ~|~ \exists q \in A, d(p, q) < \varepsilon \} = c up _{p} B _{\varepsilon} (p).

where

B _{\varepsilon} (p) is the open ball of radius \varepsilon centered at p .

The Lévy–Prokhorov metric

\pi : l{P} (M) ^{2} \to [ 0 , + inf ty) is defined by setting the distance between two probability measures \mu and \nu to be\pi (\mu, \nu) : = inf \left\{ \varepsilon > 0 ~|~ \mu(A) \leq \nu (A ^{\varepsilon} ) + \varepsilon and \nu (A) \leq \mu (A ^{\varepsilon} ) + \varepsilon forall A \in l{B}(M) \right\}.

For probability measures clearly

\pi (\mu, \nu) \le 1 .A ; either inequality implies the other, and (\bar{A}) ^{\varepsilon} = A ^{\varepsilon} , 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, \pi is a metrization of the topology of weak convergence on l{P} (M) .

• The metric space

\left( l{P} (M), \pi \right) is separable if and only if (M, d) is separable.

• If

\left( l{P} (M), \pi \right) is complete then (M, d) is complete. If all the measures in l{P} (M) have separable support, then the converse implication also holds: if (M, d) is complete then \left( l{P} (M), \pi \right) is complete. In particular, this is the case if (M, d) is separable.

• If

(M, d) is separable and complete, a subset l{K} \subseteq l{P} (M) is relatively compact if and only if its \pi -closure is \pi -compact.

• If

(M,d) is separable, then \pi (\mu , \nu ) = inf \{ \alpha (X,Y) : Law (X) = \mu , Law (Y) = \nu \} , where \alpha (X,Y) = inf \{ \varepsilon > 0 : P ( d( X ,Y ) > \varepsilon ) \leq \varepsilon \} is the Ky Fan metric.

Relation to other distances

Let

(M,d) be separable. Then\pi (\mu , \nu ) \leq \delta (\mu , \nu) , where \delta (\mu,\nu) is the total variation distance of probability measures^[1] \pi (\mu , \nu) ^{2} \leq W _{p} (\mu, \nu) ^{p} , where W _{p} is the Wasserstein metric with p\geq 1 and \mu, \nu have finite p th moment.

See also

• Lévy metric
• Prokhorov's theorem
• Tightness of measures
• Weak convergence of measures
• Wasserstein metric
• Radon distance
• Total variation distance of probability measures

References

• Book: Billingsley, Patrick . Convergence of Probability Measures . John Wiley & Sons, Inc., New York . 1999 . 0-471-19745-9 . 41238534 . registration .
  • Book: Dudley, R.M.. 1989. Real analysis and probability. Pacific Grove, Calif. : Wadsworth & Brooks/Cole. 0-534-10050-3.
• Book: Račev, Svetlozar T.. 1991. Probability metrics and the stability of stochastic models. Chichester [u.a.] : Wiley. 0-471-92877-1.

Notes and References

1. 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.