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
be a metric space with its Borel sigma algebra
. Let
denote the collection of all probability measures on the measurable space
.
For a subset
, define the ε-neighborhood of
by
![{\displaystyle A^{\varepsilon }:=\{p\in M~|~\exists q\in A,\ d(p,q)<\varepsilon \}=\bigcup _{p\in A}B_{\varepsilon }(p).}](//wikimedia.org/api/rest_v1/media/math/render/svg/ec894595bd3b8d56306f13b4c21f8ba3339abd14)
where
is the open ball of radius
centered at
.
The Lévy–Prokhorov metric :{\mathcal {P}}(M)^{2}\to [0,+\infty )}
is defined by setting the distance between two probability measures
and
to be
![{\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\}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/a0e80e4a96aa70065d8f8c7fb01b8d0142bddd69)
For probability measures clearly
.
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
is not Polish).
- If
is separable, convergence of measures in the Lévy–Prokhorov metric is equivalent to weak convergence of measures. Thus,
is a metrization of the topology of weak convergence on
.
- The metric space
is separable if and only if
is separable.
- If
is complete then
is complete. If all the measures in
have separable support, then the converse implication also holds: if
is complete then
is complete. In particular, this is the case if
is separable.
- If
is separable and complete, a subset
is relatively compact if and only if its
-closure is
-compact.
- If
is separable, then
, where
is the Ky Fan metric.[1][2]
Let
be separable. Then
, where
is the total variation distance of probability measures[3]
, where
is the Wasserstein metric with
and
have finite
th moment.[4]
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.
- Billingsley, Patrick (1999). Convergence of Probability Measures. John Wiley & Sons, Inc., New York. ISBN 0-471-19745-9. OCLC 41238534.
- Zolotarev, V.M. (2001) [1994], "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 [u.a.] : Wiley. ISBN 0-471-92877-1.