Definitions

# 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 Pierre Lévy and the Soviet mathematician Yuri Vasilevich Prokhorov; Prokhorov introduced it in 1956 as a generalization of the earlier Lévy metric.

## Definition

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

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

$A^\left\{varepsilon\right\} := \left\{ p in M | exists q in A, d\left(p, q\right) < varepsilon \right\} = bigcup_\left\{p in A\right\} B_\left\{varepsilon\right\} \left(p\right).$

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

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

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

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

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 $\left(M, d\right)$ 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 $left\left(mathcal\left\{P\right\} \left(M\right), pi right\right)$ is separable if and only if $\left(M, d\right)$ is separable.
• If $left\left(mathcal\left\{P\right\} \left(M\right), pi right\right)$ is complete then $\left(M, d\right)$ is complete. If all the measures in $mathcal\left\{P\right\} \left(M\right)$ have separable support, then the converse implication also holds: if $\left(M, d\right)$ is complete then $left\left(mathcal\left\{P\right\} \left(M\right), pi right\right)$ is complete.
• If $\left(M, d\right)$ is separable and complete, a subset $mathcal\left\{K\right\} subseteq mathcal\left\{P\right\} \left(M\right)$ is relatively compact if and only if its $pi$-closure is $pi$-compact.