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.
Let ( M , d ) be a metric space with its Borel sigma algebra B ( M ) . Let P ( M ) denote the collection of all probability measures on the measurable space ( M , B ( M ) ) .
For a subset A ⊆ M , define the ε-neighborhood of A by
A ε := { p ∈ M | ∃ q ∈ A , d ( p , q ) < ε } = ⋃ p ∈ A B ε ( p ) . where B ε ( p ) is the open ball of radius ε centered at p .
The Lévy–Prokhorov metric π : P ( M ) 2 → [ 0 , + ∞ ) is defined by setting the distance between two probability measures μ and ν to be
π ( μ , ν ) := inf { ε > 0 | μ ( A ) ≤ ν ( A ε ) + ε and ν ( A ) ≤ μ ( A ε ) + ε for all A ∈ B ( M ) } . For probability measures clearly π ( μ , ν ) ≤ 1 .
Some authors omit one of the two inequalities or choose only open or closed A ; either inequality implies the other, and ( A ¯ ) ε = A ε , but restricting to open sets may change the metric so defined (if M is not Polish).
If ( M , d ) is separable, convergence of measures in the Lévy–Prokhorov metric is equivalent to weak convergence of measures. Thus, π is a metrization of the topology of weak convergence on P ( M ) .The metric space ( P ( M ) , π ) is separable if and only if ( M , d ) is separable.If ( P ( M ) , π ) is complete then ( M , d ) is complete. If all the measures in P ( M ) have separable support, then the converse implication also holds: if ( M , d ) is complete then ( P ( M ) , π ) is complete.If ( M , d ) is separable and complete, a subset K ⊆ P ( M ) is relatively compact if and only if its π -closure is π -compact.