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.

Relation to other distances

Let be separable. Then

• , where is the total variation distance of probability measures
• , where is the Wasserstein metric with and have finite 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

Notes

References