, 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 Pierre Lévy
and the Soviet
mathematician Yuri Vasilevich Prokhorov
; Prokhorov introduced it in 1956
as a generalization of the earlier Lévy metric
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, but restricting to open/closed sets changes the metric so defined.
- 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.
- 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.
- Billingsley, Patrick (1999). Convergence of Probability Measures. John Wiley & Sons, Inc., New York.