Lévy–Prokhorov metric

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 be a metric space with its Borel sigma algebra . Let denote the collection of all probability measures on the measurable space .

For a subset , define the ε-neighborhood of by

where is the open ball of radius centered at .

The Lévy–Prokhorov metric is defined by setting the distance between two probability measures and to be

For probability measures clearly .

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

Properties

  • If is separable, convergence of measures in the Lévy–Prokhorov metric is equivalent to weak convergence of measures. Thus, is a metrization of the topology of weak convergence on .
  • The metric space is separable if and only if is separable.
  • If is complete then is complete. If all the measures in have separable support, then the converse implication also holds: if is complete then is complete. In particular, this is the case if is separable.
  • If is separable and complete, a subset is relatively compact if and only if its -closure is -compact.
  • If is separable, then , where is the Ky Fan metric.[1][2]

Relation to other distances

Let be separable. Then

  • , where is the total variation distance of probability measures[3]
  • , where is the Wasserstein metric with and have finite th moment.[4]

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

  • Billingsley, Patrick (1999). Convergence of Probability Measures. John Wiley & Sons, Inc., New York. ISBN 0-471-19745-9. OCLC 41238534.
  • Zolotarev, V.M. (2001) [1994], "Lévy–Prokhorov metric", Encyclopedia of Mathematics, EMS Press
  • Dudley, R.M. (1989). Real analysis and probability. Pacific Grove, Calif. : Wadsworth & Brooks/Cole. ISBN 0-534-10050-3.
  • Račev, Svetlozar T. (1991). Probability metrics and the stability of stochastic models. Chichester [u.a.] : Wiley. ISBN 0-471-92877-1.
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