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 [math]\displaystyle{ (M, d) }[/math] be a metric space with its Borel sigma algebra [math]\displaystyle{ \mathcal{B} (M) }[/math]. Let [math]\displaystyle{ \mathcal{P} (M) }[/math] denote the collection of all probability measures on the measurable space [math]\displaystyle{ (M, \mathcal{B} (M)) }[/math].

For a subset [math]\displaystyle{ A \subseteq M }[/math], define the ε-neighborhood of [math]\displaystyle{ A }[/math] by

[math]\displaystyle{ A^{\varepsilon} := \{ p \in M ~|~ \exists q \in A, \ d(p, q) \lt \varepsilon \} = \bigcup_{p \in A} B_{\varepsilon} (p). }[/math]

where [math]\displaystyle{ B_{\varepsilon} (p) }[/math] is the open ball of radius [math]\displaystyle{ \varepsilon }[/math] centered at [math]\displaystyle{ p }[/math].

The Lévy–Prokhorov metric [math]\displaystyle{ \pi : \mathcal{P} (M)^{2} \to [0, + \infty) }[/math] is defined by setting the distance between two probability measures [math]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \nu }[/math] to be

[math]\displaystyle{ \pi (\mu, \nu) := \inf \left\{ \varepsilon \gt 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\}. }[/math]

For probability measures clearly [math]\displaystyle{ \pi (\mu, \nu) \le 1 }[/math].

Some authors omit one of the two inequalities or choose only open or closed [math]\displaystyle{ A }[/math]; either inequality implies the other, and [math]\displaystyle{ (\bar{A})^\varepsilon = A^\varepsilon }[/math], but restricting to open sets may change the metric so defined (if [math]\displaystyle{ M }[/math] is not Polish).

Properties

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

Relation to other distances

Let [math]\displaystyle{ (M,d) }[/math] be separable. Then

• [math]\displaystyle{ \pi (\mu , \nu ) \leq \delta (\mu , \nu) }[/math], where [math]\displaystyle{ \delta (\mu,\nu) }[/math] is the total variation distance of probability measures
• [math]\displaystyle{ \pi (\mu , \nu)^2 \leq W_p (\mu, \nu)^p }[/math], where [math]\displaystyle{ W_p }[/math] is the Wasserstein metric with [math]\displaystyle{ p\geq 1 }[/math] and [math]\displaystyle{ \mu, \nu }[/math] have finite [math]\displaystyle{ p }[/math]th moment.

See also

• Lévy metric
• Prokhorov's theorem
• Tightness of measures
• Weak convergence of measures
• Wasserstein metric
• Radon distance
• Total variation distance of probability measures

References

• Billingsley, Patrick (1999). Convergence of Probability Measures. John Wiley & Sons, Inc., New York. ISBN 0-471-19745-9. OCLC 41238534. https://archive.org/details/convergenceofpro0000bill. 
• Hazewinkel, Michiel, ed. (2001), "Lévy–Prokhorov metric", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=l/l058320 
• 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.