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 (M, d) 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 A by where is the open ball of radius \varepsilon centered at p. The Lévy–Prokhorov metric is defined by setting the distance between two probability measures \mu and \nu to be For probability measures clearly. Some authors omit one of the two inequalities or choose only open or closed A; either inequality implies the other, and, but restricting to open sets may change the metric so defined (if M is not Polish).

Properties

Relation to other distances

Let (M,d) be separable. Then