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

For a subset $A\subseteq M$, define the ε-neighborhood of $A$ by

$A^{\varepsilon }:=\{p\in M~|~\exists q\in A,\ d(p,q)<\varepsilon \}=\bigcup _{p\in A}B_{\varepsilon }(p).$

where $B_{\varepsilon }(p)$ is the open ball of radius $\varepsilon$ centered at $p$.

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

$\pi (\mu ,\nu ):=\inf \left\{\varepsilon >0~|~\mu (A)\leq \nu (A^{\varepsilon })+\varepsilon \ {\text{and}}\ \nu (A)\leq \mu (A^{\varepsilon })+\varepsilon \ {\text{for all}}\ A\in {\mathcal {B}}(M)\right\}.$

For probability measures clearly $\pi (\mu ,\nu )\leq 1$.

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

Properties

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

Relation to other distances

Let $(M,d)$ be separable. Then

• $\pi (\mu ,\nu )\leq \delta (\mu ,\nu )$, where $\delta (\mu ,\nu )$ is the total variation distance of probability measures
• $\pi (\mu ,\nu )^{2}\leq W_{p}(\mu ,\nu )^{p}$, where $W_{p}$ is the Wasserstein metric with $p\geq 1$ and $\mu ,\nu$ have finite $p$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.
• 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.