Lévy–Prokhorov metric


In mathematics, the Lévy–Prokhorov metric is a metric 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.

Properties