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.
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).
- 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.
- Lévy metric
- Prokhorov's theorem
- Tightness of measures
- weak convergence of measures
- Wasserstein metric
- Radon distance
- Billingsley, Patrick (1999). Convergence of Probability Measures. John Wiley & Sons, Inc., New York. ISBN 0-471-19745-9. OCLC 41238534.
- Zolotarev, V.M. (2001) , "Lévy–Prokhorov metric", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4