Lévy–Prokhorov metric
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))} .
be a 
. Let
P
(
M
)
{\displaystyle {\mathcal {P}}(M)}
denote the collection of all 
.
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).}
, define the 
by
where B ε ( p ) {\displaystyle B_{\varepsilon }(p)} is the open ball of radius ε {\displaystyle \varepsilon } centered at p {\displaystyle p} .
is the 
centered at
p
{\displaystyle p}
.
The Lévy–Prokhorov metric π : P ( M ) 2 →
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.
, where
δ
(
μ
,
ν
)
{\displaystyle \delta (\mu ,\nu )}
is the 
, where
W
p
{\displaystyle W_{p}}
is the 
and
μ
,
ν
{\displaystyle \mu ,\nu }
have finite
p
{\displaystyle p}
See also
- Lévy metric
- Prokhorov's theorem
- Tightness of measures
- Weak convergence of measures
- Wasserstein metric
- Radon distance
- Total variation distance of probability measures
Notes
- ^ Dudley 1989, p. 322
- ^ Račev 1991, p. 159
- ^ 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.
- ^ Račev 1991, p. 175
References
- Billingsley, Patrick (1999). Convergence of Probability Measures. John Wiley & Sons, Inc., New York. ISBN 0-471-19745-9. OCLC 41238534.
- Zolotarev, V.M. (2001) , "Lévy–Prokhorov metric", Encyclopedia of Mathematics, EMS Press
- Dudley, R.M. (1989). Real analysis and probability. Pacific Grove, Calif. : Wadsworth & Brooks/Cole. ISBN 0-534-10050-3.
- Račev, Svetlozar T. (1991). Probability metrics and the stability of stochastic models. Chichester : Wiley. ISBN 0-471-92877-1.