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 $\mathcal (M)$. Let $\mathcal (M)$ denote the collection of all probability measures on the measurable space $(M, \mathcal (M))$.

For a subset $A \subseteq M$, define the ε-neighborhood of $A$ by
:$A^ := \ = \bigcup_ B_ (p).$

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

The Lévy–Prokhorov metric $\pi : \mathcal (M)^ \to$open or closed $A$; either inequality implies the other, and $(\bar)^\varepsilon = A^\varepsilon$, 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 separable space">separable, convergence of measures in the Lévy–Prokhorov metric is equivalent to metrization of the topology of weak convergence on $\mathcal (M)$.
* The metric space $\left( \mathcal (M), \pi \right)$ is separable separable space">separable if and only if $(M, d)$ is separable.
* If $\left( \mathcal (M), \pi \right)$ is complete space">complete then $(M, d)$ is complete. If all the measures in $\mathcal (M)$ have separable support (measure theory)">support