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Lévy–Prokhorov metric

From Wikipedia, the free encyclopedia

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.
  • If is separable and complete, a subset is relatively compact if and only if its -closure is -compact.

See also

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", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
This page was last edited on 10 March 2017, at 19:12
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