Lévy–Prokhorov metric

Template:Short description 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 $ℬ(M)$. Let $𝒫(M)$ denote the collection of all probability measures on the measurable space $(M,ℬ(M))$.

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

$A^{\epsilon }:=\{p\in M|\mathrm{\exists }q\in A,d(p,q)<\epsilon \}=\bigcup _{p\in A}{B}_{\epsilon }(p).$

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

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

$\pi (\mu ,\nu ):=inf\{\epsilon >0|\mu (A)\le \nu (A^{\epsilon })+\epsilon \text{and}\nu (A)\le \mu (A^{\epsilon })+\epsilon \text{for all}A\in ℬ(M)\}.$

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

Some authors omit one of the two inequalities or choose only open or closed $A$; either inequality implies the other, and $(\overline{A}{)}^{\epsilon }=A^{\epsilon }$, 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 $𝒫(M)$.
• The metric space $(𝒫(M),\pi )$ is separable if and only if $(M,d)$ is separable.
• If $(𝒫(M),\pi )$ is complete then $(M,d)$ is complete. If all the measures in $𝒫(M)$ have separable support, then the converse implication also holds: if $(M,d)$ is complete then $(𝒫(M),\pi )$ is complete. In particular, this is the case if $(M,d)$ is separable.
• If $(M,d)$ is separable and complete, a subset $𝒦\subseteq 𝒫(M)$ is relatively compact if and only if its $\pi$-closure is $\pi$-compact.
• Template:AnchorIf $(M,d)$ is separable, then $\pi (\mu ,\nu )=inf\{\alpha (X,Y):\text{Law}(X)=\mu ,\text{Law}(Y)=\nu \}$, where $\alpha (X,Y)=inf\{\epsilon >0:\mathbb{P}(d(X,Y)>\epsilon )\le \epsilon \}$ is the Ky Fan metric.

Relation to other distances

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

• $\pi (\mu ,\nu )\le \delta (\mu ,\nu )$, where $\delta (\mu ,\nu )$ is the total variation distance of probability measures
• $\pi (\mu ,\nu {)}^2\le W_p(\mu ,\nu {)}^p$, where $W_p$ is the Wasserstein metric with $p\ge 1$ 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

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