Kepler–Bouwkamp constant

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 Vasilevich 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

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

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

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

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

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, but restricting to open/closed sets changes the metric so defined.

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.
• The metric space $(\mathcal{𝒫}(M),\pi )$ is separable if and only if $(M,d)$ is separable.
• If $(\mathcal{𝒫}(M),\pi )$ 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 $(\mathcal{𝒫}(M),\pi )$ is complete.
• 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.

See also

• Lévy metric
• Wasserstein metric

References

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