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 ) {\displaystyle (M,d)} be a metric space with its Borel sigma algebra B ( M ) {\displaystyle {\mathcal {B}}(M)} . Let P ( M ) {\displaystyle {\mathcal {P}}(M)} denote the collection of all probability measures on the measurable space ( M , B ( M ) ) {\displaystyle (M,{\mathcal {B}}(M))} . For a subset A ⊆ M {\displaystyle A\subseteq M} , define the ε-neighborhood of A {\displaystyle A} by
A ε := { p ∈ M | ∃ q ∈ A , d ( p , q ) < ε } = ⋃ p ∈ A B ε ( p ) . {\displaystyle A^{\varepsilon }:=\{p\in M~|~\exists q\in A,\ d(p,q)<\varepsilon \}=\bigcup _{p\in A}B_{\varepsilon }(p).}
where B ε ( p ) {\displaystyle B_{\varepsilon }(p)} is the open ball of radius ε {\displaystyle \varepsilon } centered at p {\displaystyle p} . The Lévy–Prokhorov metric π : P ( M ) 2 → [ 0 , + ∞ ) {\displaystyle \pi :{\mathcal {P}}(M)^{2}\to [0,+\infty )} is defined by setting the distance between two probability measures μ {\displaystyle \mu } and ν {\displaystyle \nu } 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 ) } . {\displaystyle \pi (\mu ,\nu ):=\inf \left\{\varepsilon >0~|~\mu (A)\leq \nu (A^{\varepsilon })+\varepsilon \ {\text{and}}\ \nu (A)\leq \mu (A^{\varepsilon })+\varepsilon \ {\text{for all}}\ A\in {\mathcal {B}}(M)\right\}.}
For probability measures clearly π ( μ , ν ) ≤ 1 {\displaystyle \pi (\mu ,\nu )\leq 1} . Some authors omit one of the two inequalities or choose only open or closed A {\displaystyle A} ; either inequality implies the other, and ( A ¯ ) ε = A ε {\displaystyle ({\bar {A}})^{\varepsilon }=A^{\varepsilon }} , but restricting to open sets may change the metric so defined (if M {\displaystyle M} is not Polish).
… excerpt ends here. Continue reading the full article.
