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.
be a metric space with its Borel sigma algebra
denote the collection of all probability measures on the measurable space
, define the ε-neighborhood of
ε
is the open ball of radius
ε
is defined by setting the distance between two probability measures
to be For probability measures clearly
π ( μ , ν ) ≤ 1
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