# 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.

## Contents

## 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

- Lévy metric
- Prokhorov's theorem
- Tightness of measures
- weak convergence of measures
- Wasserstein metric

## 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), "Lévy–Prokhorov metric", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4