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Lévy–Prokhorov metric

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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).

Properties

  • If ( M , d ) {\displaystyle (M,d)} is separable, convergence of measures in the Lévy–Prokhorov metric is equivalent to weak convergence of measures. Thus, π {\displaystyle \pi } is a metrization of the topology of weak convergence on P ( M ) {\displaystyle {\mathcal {P}}(M)} .
  • The metric space ( P ( M ) , π ) {\displaystyle \left({\mathcal {P}}(M),\pi \right)} is separable if and only if ( M , d ) {\displaystyle (M,d)} is separable.
  • If ( P ( M ) , π ) {\displaystyle \left({\mathcal {P}}(M),\pi \right)} is complete then ( M , d ) {\displaystyle (M,d)} is complete. If all the measures in P ( M ) {\displaystyle {\mathcal {P}}(M)} have separable support, then the converse implication also holds: if ( M , d ) {\displaystyle (M,d)} is complete then ( P ( M ) , π ) {\displaystyle \left({\mathcal {P}}(M),\pi \right)} is complete. In particular, this is the case if ( M , d ) {\displaystyle (M,d)} is separable.
  • If ( M , d ) {\displaystyle (M,d)} is separable and complete, a subset K ⊆ P ( M ) {\displaystyle {\mathcal {K}}\subseteq {\mathcal {P}}(M)} is relatively compact if and only if its π {\displaystyle \pi } -closure is π {\displaystyle \pi } -compact.
  • If ( M , d ) {\displaystyle (M,d)} is separable, then π ( μ , ν ) = inf { α ( X , Y ) : Law ( X ) = μ , Law ( Y ) = ν } {\displaystyle \pi (\mu ,\nu )=\inf\{\alpha (X,Y):{\text{Law}}(X)=\mu ,{\text{Law}}(Y)=\nu \}} , where α ( X , Y ) = inf { ε > 0 : P ( d ( X , Y ) > ε ) ≤ ε } {\displaystyle \alpha (X,Y)=\inf\{\varepsilon >0:\mathbb {P} (d(X,Y)>\varepsilon )\leq \varepsilon \}} is the Ky Fan metric.

Relation to other distances

Let ( M , d ) {\displaystyle (M,d)} be separable. Then

  • π ( μ , ν ) ≤ δ ( μ , ν ) {\displaystyle \pi (\mu ,\nu )\leq \delta (\mu ,\nu )} , where δ ( μ , ν ) {\displaystyle \delta (\mu ,\nu )} is the total variation distance of probability measures
  • π ( μ , ν ) 2 ≤ W p ( μ , ν ) p {\displaystyle \pi (\mu ,\nu )^{2}\leq W_{p}(\mu ,\nu )^{p}} , where W p {\displaystyle W_{p}} is the Wasserstein metric with p ≥ 1 {\displaystyle p\geq 1} and μ , ν {\displaystyle \mu ,\nu } have finite p {\displaystyle p} th moment.

See also

Notes

  1. Dudley 1989, p. 322
  2. Račev 1991, p. 159
  3. Gibbs, Alison L.; Su, Francis Edward: On Choosing and Bounding Probability Metrics, International Statistical Review / Revue Internationale de Statistique, Vol 70 (3), pp. 419-435, Lecture Notes in Math., 2002.
  4. Račev 1991, p. 175

References

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